Chen.L;Xu.J2007a

@incollection{Chen.L;Xu.J2007a,
Abstract = {},
Author = {Long Chen and J. Xu},
Title = {A Posteriori Error Estimator by Post-processing},
Journal = {},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Chen.L;Xu.J2007

@incollection{Chen.L;Xu.J2007,
Abstract = {},
Author = {Long Chen and J. Xu},
Title = {Convergence of Adaptive Finite Element Methods},
Journal = {},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Sun.P;Russell.R;Xu.J2007

@article{Sun.P;Russell.R;Xu.J2007,
Abstract = {A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the Lp norm ($1< p \leq \infty $) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459--468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem.},
Author = {P. Sun and Robert D. Russell and J. Xu},
Title = {A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem},
Journal = {Journal of Computational Physics},
Volume = {224},
Pages = {1021--1048},
Year = {2007},
Note = {},
}

Bank.R;Xu.J;Zheng.B2007

@article{Bank.R;Xu.J;Zheng.B2007,
Abstract = {In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution $u_h$ on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree $p\geq 1$, by applying the global $L^2$ projection ($Q_h$) and a smoothing operator ($S_h$), the recovered $p$th derivatives ($S_h^m Q_h\partial^p u_h$) superconverge to the exact derivatives ($\partial^p u$). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the $(p+1)$st derivatives approximated by $\partial S_h^m Q_h\partial^p u_h$. We provide several numerical examples illustrating the effectiveness of our schemes. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence.},
Author = {R. E. Bank and J. Xu and Bin Zheng},
Title = {Superconvergent derivative recovery for Lagrange triangular elements of degree p on unstructured grids},
Journal = {SIAM J. Numerical Analysis},
Volume = {45},
Pages = {2032--2046},
Year = {2007},
Note = {},
}

Lee.Y;Wu.J;Xu.J;Zikatanov.L2007

@article{Lee.Y;Wu.J;Xu.J;Zikatanov.L2007,
Abstract = {In this paper we discuss convergence results for general (successive) subspace correction methods for solving nearly singular systems of equations. We provide parameter independent estimates under appropriate assumptions on the subspace solvers and space decompositions. The main assumption is that any component in the kernel of the singular part of the system can be decomposed into a sum of local (in each subspace) kernel components. This assumption also covers the case of "hidden" nearly singular behavior due to decreasing mesh size in the systems resulting from finite element discretizations of second order elliptic problems. To illustrate our abstract convergence framework, we analyze a multilevel method for the Neumann problem (H(grad) system), and also two-level methods for H(div) and H(curl) systems.},
Author = {Y. Lee and J. Wu and J. Xu and L. Zikatanov},
Title = {Robust subspace correction methods for nearly singular systems},
Journal = {Mathematical Models and Methods in Applied Sciences},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Xu.J;Zhu.Y2007

@article{Xu.J;Zhu.Y2007,
Abstract = {},
Author = {J. Xu and Y. Zhu},
Title = {Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients},
Journal = {Mathematical Models and Methods in Applied Sciences},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Shu.S;Babuska.I;Xu.J;Xiao.Y;Zikatanov.L2007

@techreport{Shu.S;Babuska.I;Xu.J;Xiao.Y;Zikatanov.L2007,
Abstract = {},
Author = {Shi Shu and Ivo Babuska and Jinchao Xu and Yongxiong Xiao and Ludmil Zikatanov},
Title = {Preconditioning Discrete Models of Lattice Block Materials},
Journal = {},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Chen.L;Holst.M;Xu.J2007a

@article{Chen.L;Holst.M;Xu.J2007a,
Abstract = {A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is introduced as an auxilliary problem, making it possible to study the original nonlinear equation with delta function sources. {\it A priori} error estimates for finite element approximation is obtained for the regularized Poisson-Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by {\it a posteriori} error estimate is shown to converge. The Poisson-Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann equation. Note that this article apparently gives the first convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta-function sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is one of only a handful of convergence results of this type for nonlinear problems. },
Author = {Long Chen and Michael Holst and Jinchao Xu},
Title = {The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation},
Journal = {SIAM J. Numerical Analysis},
Volume = {45},
Pages = {2298--2320},
Year = {2007},
Note = {},
}

Lee.Y;Wu.J;Xu.J2007a

@article{Lee.Y;Wu.J;Xu.J2007a,
Abstract = {This paper is devoted to the convergence rate estimate for the method of successive subspace corrections applied to symmetric and positive semidefinite (singular) problems. In a general Hilbert space setting, a convergence rate identity is obtained for the method of subspace corrections in terms of the subspace solvers. As an illustration, the new abstract theory is used to show uniform convergence of a multigrid method applied to the solution of the Laplace equation with pure Neumann boundary conditions.},
Author = {Y. Lee and J. Wu and Jinchao Xu and L. Zikatanov},
Title = {A sharp convergence estimate for the method of subspace corrections for singular systems of equations},
Journal = {Mathematics of Computation},
Volume = {},
Pages = {},
Year = {2007},
Note = {},
}

Brannick.J;Brezina.M;Falgout.R2006

@article{Brannick.J;Brezina.M;Falgout.R2006,
Abstract = {Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. Specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics.},
Author = {Brannick, J. and Brezina, M. and Falgout, R. and Manteuffel, T. and Mccormick, S. and Ruge, J. and Sheehan, B. and Xu, J. and Zikatanov, L.},
Title = {Extending the applicability of multigrid methods},
Journal = {J. Phys.: Conf. Ser.},
Volume = {46},
Pages = {443--452},
Year = {2006},
Note = {},
}

Hiptmair.R;Xu.J2006

@article{Hiptmair.R;Xu.J2006,
Abstract = { In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of $\Hcurl$- and $\Hdiv$-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by appealing to the abstract theory of auxiliary space preconditioning. The main tool are discrete analogues of so-called regular decomposition results in the function spaces H(curl) and H(div). Our preconditioner for H(curl) space is similar to an algorithm proposed in [{\sc R.~Beck}, {\em Algebraic multigrid by component splitting for edge elements on simplicial triangulations}, Techn. Report SC 99-40, ZIB, Berlin, Germany, 1999.]. },
Author = {R. Hiptmair and J. Xu},
Title = {Nodal Auxiliary Space Preconditioning in H(curl) and H(div) spaces},
Journal = {SIAM J. Numerical Analysis},
Volume = {},
Pages = {},
Year = {2006},
Note = {},
}

Mu.M;Xu.J2006

@article{Mu.M;Xu.J2006,
Abstract = {We study numerical methods for solving a coupled Stokes-Darcy problem in porous media flow applications. A two-grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions. Error estimates are derived for the proposed method. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid approach for solving multi-modeling problems. Potential extensions and future directions are discussed.},
Author = {M. Mu and J. Xu},
Title = {A Two-grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow},
Journal = {SIAM J. Numerical Analysis},
Volume = {45},
Pages = {1801--1813},
Year = {2007},
Note = {},
}

Wang.M;Xu.J2006b

@article{Wang.M;Xu.J2006b,
Abstract = {This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operator in three spatial dimensions. The newly constructed elements include two tetrahedron nonconforming finite elements and one quasi-conforming tetrahedron element. These elements are all proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element while the non-modified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special but regular grid.},
Author = {M. Wang and J. Xu},
Title = {Nonconforming Tetrahedral Finite Elements for Fourth Order Elliptic Equations},
Journal = {Mathematics of Computation},
Volume = {76},
Pages = {1--18},
Year = {2007},
Note = {},
}

Wang.M;Xu.J2006a

@article{Wang.M;Xu.J2006a,
Abstract = {This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree that provide a consistent approximation of Sobolev spaces $H^m$ in $R^n$ (with $n\ge m\ge 1$) and also a convergent (nonconforming) finite element space for $2m$-th order elliptic boundary value problems in $R^n$. This class of spaces, denoted by $M^{m}_h$, are given by piecewise polynomials with degree not greater than $m$, namely the space $P_m$. Degrees of freedom for $M^{m}_h$ in each element are given in terms of integral averages of normal derivatives of order $m-k$ on all subsimplexes of dimension $n-k$ for $1\le k\le m$. The total number of these degrees of freedom in each element amounts to $C_{n+m}^m$ which is precisely the dimension of $P_m$. One remarkable property of these sequence of spaces $M^{m}_h$ is that $\partial_i M^{m}_h\subset M^{m-1}_h$ and, furthermore, span$(\partial_1 M^{m}_h,\partial_2 M^{m}_h, \ldots, \partial_n M^{m}_h)=M^{m-1}_h$. The finite element spaces $M^{m}_h$ constructed in this paper is the only class of finite element spaces that are known and proved to be convergent for approximation of any $2m$-th order elliptic problems in any $R^n$ such that $n\ge m\ge 1$. It recovers the non-conforming linear elements for the Poisson equations ($m=1$) and the well-known Morley element for biharmonic equations ($m=2$). In order to analyze the convergence of the new class of finite element method, a general convergence theory based on a simple weak continuity assumption is also developed in this paper for nonconforming finite element methods. This new theory can be applied directly to all the simplicial and tensor-product nonconforming finite elements that are known to the authors, including the new finite element spaces proposed in this paper. For both theoretical and practical considerations, a procedure of constructing nodal basis functions of the new finite element spaces is also presented in the paper. },
Author = {M. Wang and J. Xu},
Title = {Minimal finite element spaces for $2m$-th order partial differential equations Minimal finite element spaces for $2m$-th order partial differential equations in R$^n$},
Journal = {Journal of The American Mathematical Society},
Volume = {},
Pages = {},
Year = {2006},
Note = {(submitted)},
}

He.Y;Xu.J;Zhou.A2006

@article{He.Y;Xu.J;Zhou.A2006,
Abstract = {Based on two-grid discretizations, some new local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. },
Author = {Y. He and J. Xu and A. Zhou},
Title = {Local and Parallel Finite Element Algorithms for the Stokes Problem},
Journal = {Numerische Mathematik},
Volume = {},
Pages = {},
Year = {2007},
Note = {To appear},
}

He.Y;Xu.J;Zhou.A2006a

@article{He.Y;Xu.J;Zhou.A2006a,
Abstract = {Based on two-grid discretizations, in this paper, some new local and parallel finite element algorithms are proposed and analyzed for the stationary incompressible Navier-Stokes problem. These algorithms are motivated by the observation that for a solution to the Navier-Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One major technical tool for the analysis is some local a priori error estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids.},
Author = {Y. He and J. Xu and A. Zhou},
Title = {Local and parallel finite element algorithms for the Navier-Stokes problem},
Journal = {Journal of Computational Mathematics},
Volume = {24},
Pages = {227-238},
Year = {2006},
Note = {},
}

Xue.G;Xu.J;Wang.C2006

@unpublished{Xue.G;Xu.J;Wang.C2006,
Abstract = {The traditional Newton's method requires certain smoothness of the coefficients of partial differential equations to get local convergence. In this paper, multilevel and continuation Newton's methods are developed for a two phase mixture flow model in porous media with nonlinear discontinuous degenerate diffusion coefficient arising in fuel cell applications. A ma jor finding is that the discrete algebraic equation after using linear finite element method is Lipschitz continuous. Numerical example shows the robustness of this method. },
Author = {Guangri Xue and Jinchao Xu and C. Y. Wang and R. Falgout},
Title = {Newton's method for a two phase mixture model with nonlinear discontinuous degenerate diffusion coefficient},
Journal = {},
Volume = {},
Pages = {},
Year = {2006},
Note = {(submitted)},
}

Xu.J;Zhu.Y;Zou.Q2006

@article{Xu.J;Zhu.Y;Zou.Q2006,
Abstract = {In this paper, we develop and analyze an adaptive finite volume algorithm for second order elliptic boundary value problems. We first derive a residual type a posteriori error estimator, and then establish upper bounds and lower bounds in comparison with the exact error. Using certain relationship between finite element and finite volume local stiffness matrix for the Poisson equation, we establish the discrete local lower bound of the error between solutions on two successive refinements. After proving a number of additional technique results including the quasi-orthogonality for two different finite volume solutions, we finally obtain the error reduction and convergence of the adaptive finite volume method.},
Author = {J. Xu and Y. Zhu and Q. Zou},
Title = {New Adaptive Finite Volume Methods and Convergence Analysis},
Journal = {Numerische Mathematik},
Volume = {},
Pages = {},
Year = {2006},
Note = {(submitted)},
}

Shu.S;Sun.D;Xu.J2006

@article{Shu.S;Sun.D;Xu.J2006,
Abstract = { In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method. },
Author = {S. Shu and D. Sun and J. Xu},
Title = {An algebraic multigrid method for higher order finite element discretizations},
Journal = {Computing},
Volume = {77},
Pages = {347--377},
Year = {2006},
Note = {},
}

Caflisch.R;Lee.Y;Shu.S2006

@article{Caflisch.R;Lee.Y;Shu.S2006,
Abstract = {We apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems, recently introduced by Schindler et al. [Phys. Rev. B 67, 075316 (2003)]. The discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth, but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge. In this paper, we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model. Using this method, simulations are performed for several representative physical problems, including an infinite periodic step train, a layered nanocrystal, and a system of quantum dots. The results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties, regardless of the problem size, the geom- etry and the physical parameters. The effects of substrate depth and of invariance due to traction-free boundary conditions are assessed. For a system of quantum dots, the simulated strain energy density supports the observations that trench for- mation near the dots provides strain relief. },
Author = {R. Caflisch and Y. Lee and S. Shu and Y. Xiao and J. Xu},
Title = {An application of multigrid methods for a discrete elastic model for epitaxial systems},
Journal = {Journal of Computational Physics},
Volume = {219},
Pages = {697--714},
Year = {2006},
Note = {},
}

Wang.M;Shi.Z;Xu.J2006c

@article{Wang.M;Shi.Z;Xu.J2006c,
Abstract = {In this paper, a new class of Zienkiewicz-type non-conforming finite element, in n spatial dimensions with $n\geq 2$, is proposed. The new finite element is proved to be convergent for the biharmonic equation.},
Author = {M. Wang and Z. Shi and J. Xu},
Title = {A New Class of Zienkiewicz-Type Non-conforming Element in Any Dimensions},
Journal = {Numerische Mathematik},
Volume = {106},
Pages = {335--347},
Year = {2007},
Note = {},
}

Lee.Y;Wu.J;Xu.J2006

@unpublished{Lee.Y;Wu.J;Xu.J2006,
Abstract = {In this paper, an abstract convergence theory for the general (successive) subspace correction methods is presented for nearly singular system of equations with a small positive parameter $\epsilon$. It is shown that the successive subspace correction methods for the augmented problem is equivalent to some successive subspace correction method for the original problem. We provide the convergence rate estimates for the general subspace correction methods and get the $\epsilon$-independent convergence under the additional assumptions on the subspace solvers and space decomposition. The main abstract assumption ({\bf A1}) implies that the kernel functions can be decomposed into a sum of kernel functions in subspaces. For an illustration, our abstract framework is applied for convergence rate analysis of subspace correction methods for $H(\grad)$, $H(\div)$ and $H(\curl)$ systems. },
Author = {Y. Lee and J. Wu and J. Xu and L. Zikatanov},
Title = {Robust subspace correction methods for nearly singular systems},
Journal = {},
Volume = {},
Pages = {},
Year = {2006},
Note = {(in preparation)},
}

Wu.J;Xu.J;Zou.H2006

@article{Wu.J;Xu.J;Zou.H2006,
Abstract = {In this paper, we shall establish the well-posedness of a mathematical model for a special class of electrochemical power device -- lithium-ion battery. The underlying partial differential equations in the model involve a (mix and fully) coupled system of quasi-linear elliptic and parabolic equations. By exploring some special structure, we are able to adopt the well-known Nash-Moser-DeGiorgi boot strap to establish suitable a priori supremum estimates for the electric potentials. Using the supremum estimates, we apply the Leray-Schauder theory to establish the existence and uniqueness of a subsystem of elliptic equations that describe the electric potentials in the model. We then employ a Schauder fix point theorem to obtain the local (in time) existence for the whole model. We also consider the global existence of a modified 1-d governing system under additional assumptions. In particular, we are able to derive uniform a priori estimates depending only on the existence time $T$, including the supremum estimates for electric potentials and growth and decay estimates for the concentration $c$. Using the uniform estimates, we prove that the modified system has a solution for all time $t>0$. },
Author = {J. Wu and J. Xu and H. Zou},
Title = {On the well posedness of mathematical model for Lithium-Ion battery systems},
Journal = {Methods and Applications of Analysis},
Volume = {},
Pages = {},
Year = {2006},
Note = {(to appear))},
}

Lee.Y;Wu.J;Xu.J2006a

@article{Lee.Y;Wu.J;Xu.J2006a,
Abstract = {Necessary and sufficient conditions for the energy norm convergence of the classical iterative methods for semi-definite linear systems are obtained in this paper. These new conditions generalize the classic notion of the P-regularity introduced by H. Keller (1965).},
Author = {Y. Lee and J. Wu and J. Xu and L. Zikatanov},
Title = {On the convergence of iterative methods for semidefinite linear systems},
Journal = {SIAM J. on Matrix Analysis},
Volume = {28},
Pages = {634--641},
Year = {2006},
Note = {},
}

Shu.S;Babuska.I;Xian.Y2006

@unpublished{Shu.S;Babuska.I;Xian.Y2006,
Abstract = {In this paper, we construct and analyze a block preconditioned conjugate gradient(BPCG) method and a class of algebraic multigrid(AMG) methods applied to the discrete mathematical models for lattice block materials. Numerical experiments show that the constructed AMG methods converge uniformly with respect to the size of problem and also to some crucial parameters. Such a uniform convergence of the BPCG algorithm is further theoretically justified by analyzing the underlying continuous models(that are recovered by a close inspection of the discrete system) for square lattice block materials without and with diagonals. },
Author = {S. Shu and I. Babuska and Y. Xian and J. Xu and L. Zikatanov},
Title = {Algebraic Multigrid Methods and Preconditioned Conjugate Gradient Algorithm for Lattice Block Materials Models},
Journal = {},
Volume = {},
Pages = {},
Year = {2006},
Note = {(in preparation)},
}

Wang.M;Shi.Z;Xu.J2006

@article{Wang.M;Shi.Z;Xu.J2006,
Abstract = {In this paper, three n-rectangle nonconforming elements are proposed with $n \ge 3$. They are the extensions of well-known Morley element, Adini element and Bogner- Fox-Schmit element in two spatial dimensions to any higher dimensions respectively. These elements are all proved to be convergent for a model biharmonic equation in n dimensions.},
Author = {M. Wang and Z. Shi and J. Xu},
Title = {Some n-Rectangle Nonconforming Elements for Fourth Order Elliptic Equations},
Journal = {Journal of Computational Mathematics},
Volume = {25},
Pages = {408--420},
Year = {2007},
Note = {(to appear)},
}

Wang.M;Xu.J;Hu.Y2006

@article{Wang.M;Xu.J;Hu.Y2006,
Abstract = {This paper propeses a modified Morley element method for a fourth order elliptic singular perturbation problem. The method also uses Morley element or rectangle Morley element, but linear or bilinear approximation of finite element functions is used in the lower part of the bilinear form. It is shown that the modified method converges uniformly in the perturbation parameter.},
Author = {M. Wang and J. Xu and Y. Hu},
Title = {Modified Morley element method for a fourth elliptic singular perturbation problem},
Journal = {J. Comp. Math},
Volume = {24},
Pages = {113-120},
Year = {2006},
Note = {},
}

Shu.S;Xu.J;Yang.Y2006

@article{Shu.S;Xu.J;Yang.Y2006,
Abstract = { In this paper, we design and analyse an algebaic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eleminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid method for more complicated problems such as unstructured grids, but, because of the speciality of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid methods. },
Author = {S. Shu and J. Xu and Y. Yang and H. Yu},
Title = {An algebraic multigrid method for finite element systems on criss-cross grids},
Journal = {Advances in Comp. Math.},
Volume = {25},
Pages = {287--304},
Year = {2006},
Note = {(to appear)},
}

Jin.J;Shu.S;Xu.J2006

@article{Jin.J;Shu.S;Xu.J2006,
Abstract = {In this paper, we propose a two-grid finite element method for solving coupled partial differential equations, e.g., the Schr\"{o}dinger type equation. With this method, the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid. It is shown, both theoretically and numerically, that the resulting solution still achieves asymptotically optimal accuracy. },
Author = {J. Jin and S. Shu and J. Xu},
Title = {A two-grid discretization method for decoupling systems of partial differential equations},
Journal = {Mathemathics of Computation},
Volume = {75},
Pages = {1617--1626},
Year = {2006},
Note = {(to appear)},
}

F.Sopuerta.C;Sun.P;Laguna.P2006

@article{F.Sopuerta.C;Sun.P;Laguna.P2006,
Abstract = {Extreme-mass-ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on a posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.},
Author = {C. F.Sopuerta and P. Sun and P. Laguna and J. Xu},
Title = {A toy model for testing finite element methods to simulate extreme-mass-ration binary systems},
Journal = {Class. Quantum Gravity},
Volume = {23},
Pages = {251-285},
Year = {2006},
Note = {},
}

Lee.Y;Xu.J2006

@article{Lee.Y;Xu.J2006,
Abstract = {We propose a class of new discretization schemes for solving the rate-type non-Newtonian constitutive equations. The so-called conformation tensor has been known to be symmetric and positive definite in a large class of constitutive equations. Preserving such a positivity property on the discrete level is believed to be crucially important but difficult. High Weissenberg number problems on numerical instabilities have been often associated with this issue. In this paper, we present various discretization schemes that preserve the positive-definiteness of the conformation tensor regardless of the time and spatial resolutions. Moreover, the robustness of the algorithm has been also demonstrated by the stability analysis using the discrete analogue of energy estimates. New schemes presented in this paper are constructed based upon the newly discovered relationship between the rate-type constitutive equations and the symmetric matrix Riccati differential equations.},
Author = {Y. Lee and J. Xu},
Title = {New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models},
Journal = {Comput. Methods Appl. Mech. Engrg.},
Volume = {195},
Pages = {1180-1206},
Year = {2006},
Note = {},
}

Chen.L;Sun.P;Xu.J2006

@article{Chen.L;Sun.P;Xu.J2006,
Abstract = {In this paper, we present a new optimal interpolation error estimate in $L^p$ norm ($1\leq p\leq \infty$) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the view points of minimizing interpolation error in the $L^p$ norm. Some numerical examples are also given to support the theoretical estimates.},
Author = {Long Chen and Pengtao Sun and Jinchao Xu},
Title = {Optimal anisotropic simplicial meshes for minimizing interpolation errors in ${L}^p$-norm},
Journal = {Mathematics of Computation},
Volume = {76},
Pages = {179--204},
Year = {2007},
Note = {},
}

Chen.L;Holst.M;Xu.J2006

@article{Chen.L;Holst.M;Xu.J2006,
Abstract = {The convergence and optimality of an adaptive mixed finite element methods for elliptic partial differential equations is established in this paper. The algorithm consists of two steps. An approximation subroutine is first called to approximate the data and then local refinement based on a posteriori error estimate is applied to the equation with piecewise constant data. },
Author = {Long Chen and Michael Holst and Jinchao Xu},
Title = {Convergence and Optimality of Adaptive Mixed Finite Element Methods},
Journal = {Mathematics of Computation},
Volume = {},
Pages = {},
Year = {2007},
Note = {(submitted)},
}

Xu.J;Zou.Q2006

@article{Xu.J;Zou.Q2006,
Abstract = {In this paper, we analyze the convergence error of the widely used finite volume methods. Regarding finite volume methods as special Petrov-Galerkin methods, we find that the convergence error analysis can be reduced to the proof of the inf-sup conditions for specific finite volume schemes. For linear finite volume methods in any arbitrary dimensional space, we prove the inf-sup condition by using some equivalence between the linear finite volume stiff matrix and that of the linear finite element method. For quadratic finite volume methods in 2D, we prove the inf-sup condition by analyzing the property of each element stiff matrix. We also obtain a super-convergence result for linear finite volume method by a simple proof.},
Author = {Jinchao Xu and Qingsong Zou},
Title = {Analysis of Linear and Quadratic Finite Volume Methods for Elliptic Equations},
Journal = {Preprint},
Volume = {},
Pages = {},
Year = {2006},
Note = {},
}

Wang.M;Xu.J2006

@article{Wang.M;Xu.J2006,
Abstract = {In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general n-dimensional Morley element consists of all quadratic polynomials defined on each n-simplex with degrees of freedom given by the integral average of the normal derivative on each (n-1)-subsimplex and the integral average of the function value on each (n-2)-subsimplex. Explicit expressions of nodal basis functions are also obtained for this element on general n-simplicial grids. Convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation.},
Author = {Ming Wang and Xu, Jinchao},
Title = {The Morley element for fourth order elliptic equations in any dimensions},
Journal = {Numerische Mathematik},
Volume = {103},
Pages = {155--169},
Year = {2006},
Note = {},
}

Yin.S;Zhang.X;Zhan.C2005

@article{Yin.S;Zhang.X;Zhan.C2005,
Abstract = {One of the biggest problems of heart failure is the heart's inability to effectively pump blood to meet the body's demands, which may be caused by disease-induced alterations in contraction properties (such as contractile force and Young's modulus). Thus, it is very important to measure contractile properties at single cardiac myocyte level that can lay the foundation for quantitatively understanding the mechanism of heart failure and understanding molecular alterations in diseased heart cells. In this article, we report a novel single cardiac myocyte contractile force measurement technique based on moving a magnetic bead. The measuring system is mainly composed of 1), a high-power inverted microscope with video output and edge detection; and 2), a moving magnetic bead based magnetic force loading module. The main measurement procedures are as follows: 1), record maximal displacement of single cardiac myocyte during contraction; 2), attach a magnetic bead on one end of the myocyte that will move with myocyte during the contraction; 3), repeat step 1 and record contraction processes under different magnitudes of magnetic force loading by adjusting the magnetic field applied on the magnetic bead; and 4), derive the myocyte contractile force base on the maximal displacement of cell contraction and magnetic loading force. The major advantages of this unique approach are: 1), measuring the force without direct connections to the cell specimen (i.e., ``remote sensing'', a noninvasive/minimally invasive approach); 2), high sensitivity and large dynamic range (force measurement range: from pico Newton to micro Newton); 3), a convenient and cost-effective approach; and 4), more importantly, it can be used to study the contractile properties of heart cells under different levels of external loading forces by adjusting the magnitude of applied magnetic field, which is very important for studying disease induced alterations in contraction properties. Experimental results demonstrated the feasibility of proposed approach.},
Author = {S. Yin and X. Zhang and C. Zhan and J. Wu and J. Cheung and J. Xu},
Title = {Measuring single cardiac myocyte contractile force via moving a magnetic bead},
Journal = {Biophysical Journal},
Volume = {},
Pages = {1489-1495},
Year = {2005},
Note = {},
}

Bacuta.C;Chen.J;Huang.Y2005

@article{Bacuta.C;Chen.J;Huang.Y2005,
Abstract = {We consider the Stokes Problem on a plane polygonal domain $\Omega\in R^2$. We propose a finite element method for overlapping or nonmatching grids for the Stokes Problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.},
Author = {C. Bacuta and J. Chen and Y. Huang and J. Xu and L. Zikatanov},
Title = {Partition of unity method on non-matching grids for the Stokes problem},
Journal = {Journal of Numerical Mathematics},
Volume = {13},
Pages = {157-169},
Year = {2005},
Note = {},
}

Chen.L;Wang.J;Xu.J2005

@article{Chen.L;Wang.J;Xu.J2005,
Abstract = {In many application domains involving shapes and curves, polygonal curve simplification is an important part of the computer analysis processes. In this work, we have developed asymptotically optimal and linear-time algorithms to approximate a polygonal curve by another polygonal curve whose vertices are a subset of the vertices of the original one. The algorithm developed in this paper can be applied to a vector map data reduction in geographical information system especially for large-scale data. The error of the approximation is measured by the area of the domain bounded by the two polygonal curves. Based on the equidistribution principle and local refinement/coarsening strategy, an efficient},
Author = {Long Chen and James Z. Wang and Jinchao Xu},
Title = {Asymptotically Optimal and Linear-time Algorithm for Polygonal Curve Simplification},
Journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
Volume = {},
Pages = {},
Year = {2005},
Note = {},
}

Chen.L;Xu.J2005a

@article{Chen.L;Xu.J2005a,
Abstract = {The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that $\|u-u_{N}\|_{\infty}\leq C\inf_{v_{N}\in V^{N}}\|u-v_{N}\|_{\infty},$ where $u_{N}$ is the SDFEM approximation in linear finite element space $V^{N}$ of the exact solution $u$. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered.},
Author = {Long Chen and Jinchao Xu},
Title = {Stability and accuracy of adapted finite element methods for singularly perturbed problems},
Journal = {Technique Report, Department of Mathematics, The Pennsylvania State University},
Volume = {},
Pages = {},
Year = {2005},
Note = {},
}

Chen.L;Xu.J2005

@inproceedings{Chen.L;Xu.J2005,
Abstract = {The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense that $\|u-u_{h}\|_{\infty}\leq C\inf_{v_{h}\in V^{h}}\|u-v_{h}\|_{\infty},$ where $u_{h}$ is the SDFEM approximation of the exact solution $u$ and $V_{h}$ is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.},
Author = {Long Chen and Jinchao Xu},
Title = {An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem},
Journal = {},
Volume = {383},
Pages = {236--246},
Year = {2005},
Note = {},
}

Huang.Y;Xu.J2005

@article{Huang.Y;Xu.J2005,
Abstract = {Superconvergence estimates are studied in this paper on quadratic finite element discretization for second order elliptic boundary value problems on mildly structured triangular meshes. For a large class of practically useful grids, the finite element solution uh is proven to be superclose to the interpolant uI and as a result a postprocessing gradient recovery scheme for uh can be devised. The analysis is based on a number of carefully derived identities. In addition to its own theoretical interests, the result in this paper can be used for deriving asymptotically exact error estimators for quadratic finite element discretization.},
Author = {Y. Q. Huang and J. Xu},
Title = {Superconvergence for quadratic triangular finite elements on mildly structured grids},
Journal = {Preprint},
Volume = {},
Pages = {},
Year = {2005},
Note = {},
}

Chen.L;Sun.P;Xu.J2004

@inproceedings{Chen.L;Sun.P;Xu.J2004,
Abstract = {A multilevel homotopic adaptive methods is presented in this paper for convection dominated problems. By the homotopy method with respect to the diffusion parameter, the grid are iteratively adapted to better approximate the solution. Some new theoretic results and practical techniques for the grid adaptation are presented. Numerical experiments show that a standard finite element scheme based on this properly adapted grid works in a robust and efficient manner.},
Author = {Long Chen and Pengtao Sun and Jinchao Xu},
Title = {Multilevel Homotopic Adaptive Finite Element Methods for Convection Dominated Problems},
Journal = {},
Volume = {},
Pages = {459--468},
Year = {2004},
Note = {},
}

Chen.L;Xu.J2004

@article{Chen.L;Xu.J2004,
Abstract = {The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in $L^p$-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function $\|\mathbf x\|^2$ among all the triangulations with a given set of vertices. For a more general function, a function-dependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-dependent optimal Delaunay triangulation is proved to exist for any given convex continuous function. On an optimal Delaunay triangulation associated with $f$, it is proved that $\nabla f$ at the interior vertices can be exactly recovered by the function values on its neighboring vertices. Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal. },
Author = {Long Chen and Jinchao Xu},
Title = {Optimal {Delaunay} triangulations},
Journal = {Journal of Computational Mathematics},
Volume = {22(2)},
Pages = {299-308},
Year = {2004},
Note = {},
}

Xu.J;Zikatanov.L2004

@article{Xu.J;Zikatanov.L2004,
Abstract = {This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then prove3d to be locally harmonic functions on each coarse grid element. Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic multigrid method, the energy minimizing basis may also be applied for numerical homogenization.},
Author = {Jinchao Xu and L. Zikatanov},
Title = {On An Energy Minimizing Basis in Algebraic Multigrid Methods},
Journal = {Computing and Visualization in Science},
Volume = {7},
Pages = {121-127},
Year = {2004},
Note = {},
}

Wu.J;Zou.H;Xu.J2004

@article{Wu.J;Zou.H;Xu.J2004,
Abstract = {In this paper, we shall establish the well-posedness of a mathematical model for a special class of electrochemical power device -- lithium-ion battery. The underlying partial differential equations in the model involve a (mix and fully) coupled system of quasi-linear elliptic and parabolic equations. By exploring some special structure, we are able to adopt the well-known Nash-Moser- DeGiorgi boot strap to establish suitable supremum a priori estimates for the electric potentials. Using the supremum estimates, we apply the Leray-Schauder theory to establish the existence and uniqueness of a subsystem of elliptic equations that describe the electric potentials in the model. We then employ a Schauder fix point theorem to obtain the local (in time) existence for the whole model. We also consider the global existence of a modified 1-d governing system under additional assumptions. In particular, we are able to derive uniform a priori estimates depending only on the existence time T , including the supremum estimates for electric potentials and growth and decay estimates for the concentration c. Utilizing the uniform estimates, we prove that the modified system has a solution for all time t > 0.},
Author = {J. Wu and H. Zou and Jinchao Xu},
Title = {On the well-posedness of a mathematical model for Lithium-Ion battery systems},
Journal = {submitted to SIAM Journal on of Math. Anal.},
Volume = {},
Pages = {},
Year = {2004},
Note = {},
}

Lee.Y;Xu.J2004

@article{Lee.Y;Xu.J2004,
Abstract = {We propose new time integrating schemes for solving the rate-type non-Newtonian constitutive equations. It has been known that most constitutive equations have certain constraints on the conformation tensor, namely the positive definiteness, however, attentions have not been paid much to keeping positivity of the conformation tensor in the discrete sense. High Weissenberg number problems have been often associated with this issue. In this paper, we present various discretization schemes which preserves the positive-definiteness of the conformation tensor regardless of the time and spatial resolutions. A crucial observation that the constitutive equations are closely related to the Riccati Differential Equation has been made in this paper.},
Author = {Y. Lee and Jinchao Xu},
Title = {Positively Preserving Schemes for the Rate-Type Non-Newtonian Fluids},
Journal = {Submitted},
Volume = {},
Pages = {},
Year = {2004},
Note = {},
}

Kim.H;Xu.J;Zikatanov.L2004

@article{Kim.H;Xu.J;Zikatanov.L2004,
Abstract = {In this paper we construct a class of multigrid methods for convection-diffusion problems. These methods are convergent without imposing any constraint on the coarsest grid mesh size. The proposed algorithms use first order stable monotone schemes to precondition the second order standard Galerkin finite element discretization. To speed up the solution process of the lower order schemes, cross-wind-block reordering of the unknowns is applied. A V-cycle iteration, based on these algorithms, is then used as a preconditioner in GMRES. The numerical examples show that the convergence of the preconditioned method is uniform.},
Author = {H. Kim and Jinchao Xu and L. Zikatanov},
Title = {Uniformly convergent multigrid methods for convection diffusion problems without any constraint on coarse grids},
Journal = {Advances in Comp. Math.},
Volume = {20},
Pages = {385--399},
Year = {2004},
Note = {},
}

Huang.Y;Xu.J2003

@article{Huang.Y;Xu.J2003,
Abstract = {In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.},
Author = {Y. Huang and Jinchao Xu},
Title = {A conforming finite element method for overlapping and nonmatching grids},
Journal = {Mathematics of Computation},
Volume = {72},
Pages = {1057--1066},
Year = {2003},
Note = {},
}

Bacuta.C;Bramble.J;Xu.J2003

@article{Bacuta.C;Bramble.J;Xu.J2003,
Abstract = {We consider the model Dirichlet problem for Poisson's equation on a plane polygonal convex domain ? with data f in a space smoother than $L^ 2$ . The regularity of the problem depends on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates including the critical case. As a consequence, sharp error estimates for corresponding discrete problems are proved. Some classical shift estimates are also proved using the tools of interpolation theory and mutilevel approximation theory. The results can be extended to a large class of elliptic boundary value problems.},
Author = {Constantin Bacuta and James H. Bramble and Jinchao Xu},
Title = {Regularity Estimates for elliptic boundary value problems with smooth data on polygonal domains},
Journal = {Numerische Mathematik},
Volume = {11},
Pages = {75--94},
Year = {2003},
Note = {},
}

Xu.J;Zikatanov.L2003b

@misc{Xu.J;Zikatanov.L2003b,
Abstract = {},
Author = {Jingchao Xu and Ludmil Zikatanov},
Title = {AMG and construction of coarse grids and related stuff},
Journal = {},
Volume = {},
Pages = {},
Year = {2003},
Note = {},
}

Xu.J;Zikatanov.L2003a

@article{Xu.J;Zikatanov.L2003a,
Abstract = {Some observations are made on abstract error estimates for Galerkin approximations based on Babuška-Brezzi conditions. A basic error estimate due to Babuška is sharpened by means of an identity that $\|P\| = \|I - P\|$ for any nontrivial idempotent operator P. Some remarks are also made on the Brezzi's theory for mixed variational problems and their Galerkin approximations.},
Author = {Jinchao Xu and L. Zikatanov},
Title = {Some Observations on {Babuška} and {Brezzi} Theories},
Journal = {Numerische Mathematik},
Volume = {94},
Pages = {195-202},
Year = {2003},
Note = {},
}

Xu.J;Zikatanov.L2003

@inproceedings{Xu.J;Zikatanov.L2003,
Abstract = {This paper reports investigations on how multigrid methods can be applied for the solution of some generalized finite element methods basd on the partition of unity technique. One feature of the generalized finite element method is that the underlying algebraic system is often singular due to the overlapping from the partition of unity. While standard iterative methods such as the conjugate gradient method, Jacobi, Gauss-Seidel methods, multigrid methods and domain decompsition methods are still convergenct for this type of singular systems, we observe that a standard multigrid method does not converge uniformly with respect to mesh parameters. Using a simple model problem, we will carefully investigate why these method do not work. We will then propose a multigrid method that does converges uniformly as in the standard finite element method.},
Author = {Jinchao Xu and L. Zikatanov},
Title = {On Multigrid Methods for Generalized Finite Element Methods},
Journal = {Lect. Notes Comput. Sci. Eng.},
Volume = {26},
Pages = {401-418},
Year = {2003},
Note = {},
}

Xu.J;Zhang.Z2003

@article{Xu.J;Zhang.Z2003,
Abstract = {Some recovery type error estimators for linear finite elements are analyzed under $O(h^{1+\alpha}) (\alpha > 0)$ regular grids. Superconvergence of order $O(h^{1+\rho}) (0 < \rho\leq \alpha)$ is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.},
Author = {Jinchao Xu and Z. M. Zhang},
Title = {Analysis of recovery type a posteriori error estimators for mildly structured grids},
Journal = {Mathematics of Computation},
Volume = {73},
Pages = {1139-1152},
Year = {2004},
Note = {},
}

Kim.H;Xu.J;Zikatanov.L2003

@article{Kim.H;Xu.J;Zikatanov.L2003,
Abstract = {This paper proposes practical and robust multigrid methods for convection diffusion problems for unstructured grids. We design new coarsening techniques on unstructured grids. The idea is to use the matching technique in order to define proper coarse space. Such a approach is based on the graph corresponding to the stiffness matrix, and is purely algebraic. Thus, there is no need of any geometrical information of the grid. We test several convection-diffusion equations with coefficients having large jumps and make the comparison with other available methods. And we conclude that the designed method is robust with respect to the jumps},
Author = {H. Kim and Jinchao Xu and L. Zikatanov},
Title = {A multigrid method based matching in graph for convection diffusion equations},
Journal = {Num. Lin. Alg. and Appl.},
Volume = {10},
Pages = {181--195},
Year = {2003},
Note = {},
}

Bank.R;Xu.J2003

@article{Bank.R;Xu.J2003,
Abstract = {In Part I of this work [SIAM Journal on Numer. Anal. , 41 (2003), pp. 2294--2312], we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In this work, we consider superconvergence for general unstructured but shape regular meshes. We develop a postprocessing gradient recovery scheme for the finite element solution uh, inspired in part by the smoothing iteration of the multigrid method. This recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact. Next, we use the superconvergent gradient to approximate the Hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms. We provide several numerical examples illustrating the effectiveness of our procedures.},
Author = {R. E. Bank and Jinchao Xu},
Title = {Asymptotically Exact A Posteriori Error Estimators, {P}art {II}: General Unstructured Grids},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {41},
Pages = {2313-2332},
Year = {2003},
Note = {},
}

Bank.R;Xu.J2003a

@article{Bank.R;Xu.J2003a,
Abstract = {In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant $u_I$. We then analyze a postprocessing gradient recovery scheme, showing that $Q_h\nabla u_h$ is a superconvergent approximation to $\nabla u$. Here Qh is the global L2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators.},
Author = {R. E. Bank and Jinchao Xu},
Title = {Asymptotically Exact A Posteriori Error Estimators, {P}art {I}: Grids with Superconvergence},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {41},
Pages = {2294-2312},
Year = {2003},
Note = {},
}

Bacuta.C;Xu.J2003

@inproceedings{Bacuta.C;Xu.J2003,
Abstract = {We consider the Stokes Problem on a plane polygonal domain ? R ^2 . We propose a finite element method for overlapping or nonmatching grids for the Stokes Problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.},
Author = {C. Bacuta and Jinchao Xu},
Title = {Partition of Unity Method for Stokes Problem on Nonmatching Grids},
Journal = {},
Volume = {},
Pages = {},
Year = {2003},
Note = {},
}

Xu.J;Zikatanov.L2002a

@article{Xu.J;Zikatanov.L2002a,
Abstract = {The method of alternating pro jections and the method of subspace corrections are general iterative methods that have a variety of applications. The method of alternating pro jections, first proposed by von Neumann (1933) (see [31]), is an algorithm for finding the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. The method of subspace corrections, an abstraction of general linear iterative methods such as multigrid and domain decomposition methods, is an algorithm for finding the solution of a linear system of equations. In this paper, we shall study these two methods in a Hilbert space setting and in particular present a new identity for the product of nonexpansive operators that gives a sharpest possible estimate of the convergence rate of these methods.},
Author = {Jinchao Xu and L. Zikatanov},
Title = {The Method of Alternating Projections and the Method of Subspace Corrections in {H}ilbert Space},
Journal = {Journal of The American Mathematical Society},
Volume = {15},
Pages = {573--597},
Year = {2002},
Note = {},
}

Bacuta.C;Bramble.J;Xu.J2002

@article{Bacuta.C;Bramble.J;Xu.J2002,
Abstract = {We consider the Dirichlet problem for Poisson's equation on a nonconvex plane polygonal domain $\d$. New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems. Some new sharp finite element error estimates are deduced.},
Author = {Constantin Bacuta and James H. Bramble and Jinchao Xu},
Title = {Regularity Estimates for elliptic boundary value problems in Besov spaes},
Journal = {Mathematics of Computation},
Volume = {72},
Pages = {1577--1595},
Year = {2002},
Note = {},
}

Xu.J;Zhou.A2002a

@article{Xu.J;Zhou.A2002a,
Abstract = {},
Author = {Jinchao Xu and A. Zhou},
Title = {Some multiscale methods for partial differential equations},
Journal = {Contemporary Mathematics},
Volume = {306},
Pages = {1--27},
Year = {2002},
Note = {},
}

Xu.J;Zhou.A2002

@article{Xu.J;Zhou.A2002,
Abstract = {Some new local and parallel finite element algorithms are proposed and analyzed in this paper for eigenvalue problems. With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraic systems on fine grid by using some local and parallel procedure. A theoretical tool for analyzing these algorithms is some local error estimate that is also obtained in this paper for finite element approximations of eigenvectors on general shape-regular grids.},
Author = {Jinchao Xu and A. Zhou},
Title = {Local and parallel finite element algorithms for eigenvalue problems},
Journal = {Acta Mathematicae Applicae},
Volume = {18},
Pages = {185--200},
Year = {2002},
Note = {},
}

Wu.J;Xu.J2002

@inproceedings{Wu.J;Xu.J2002,
Abstract = {},
Author = {J. Wu and Jinchao Xu},
Title = {Mathematical modelling and numerical simulations on electrochemical devices},
Journal = {},
Volume = {},
Pages = {},
Year = {2002},
Note = {},
}

Tai.X;Xu.J2002

@article{Tai.X;Xu.J2002,
Abstract = {This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.},
Author = {Tai, X. and Xu, J.},
Title = {Global and uniform convergence of subspace correction methods for some convex optimization problems},
Journal = {Mathematics of Computation},
Volume = {71},
Pages = {105--124},
Year = {2002},
Note = {},
}

Lee.Y;Wu.J;Xu.J2002

@inproceedings{Lee.Y;Wu.J;Xu.J2002,
Abstract = {The method of successive subspace corrections, an abstraction of general iterative methods such as multigrid and Multiplicative Schwarz methods, is an algorithm for finding the solution of a linear system of equations. In this paper, we shall study in particular, Multiplicative Schwarz methods in a Hilbert space framework and present a sharp result on the convergence of the methods for singular system of equations. For the symmetric positive definite (SPD) problems, a variety of literatures on the convergence analysis are available. Among others, we would like to refer to the upcoming paper by Xu and Zikatanov (Refer to [3]). In [3], the convergence rate of the method of subspace corrections has been beautifully established by introducing a new identity for the product of nonexpansive operators. The main result in this paper is in that we obtained an appropriate identity for the non- SPD problems, which is suitably applied to devise or improve algorithms for singular and especially nearly singular system of equations. The related results and the corresponding estimate of the convergence rate of multigrid methods for singular system of equations shall be reported in the forthcoming paper.},
Author = {Y. Lee and J. Wu and Jinchao Xu and L. Zikatanov},
Title = {Successive Subspace Correction method for Singular System of Equations},
Journal = {},
Volume = {},
Pages = {},
Year = {2002},
Note = {},
}

Xu.J2001

@article{Xu.J2001,
Abstract = {This paper gives an overview for the method of subspace corrections. The method is first motivated by a discussion on the local behavior of high frequency components in a solution to an elliptic problem. A simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more details. Brief discussions are also given to some nonlinear examples including eigenvalue problems, obstacle problems and liquid crystal modelings. The relationship between the method of subspace correction and the method of alternating projects is observed and discussed.},
Author = {Jinchao Xu},
Title = {The method of subspace corrections},
Journal = {J. Comp. Appl. Math.},
Volume = {128},
Pages = {335--362},
Year = {2001},
Note = {},
}

Xu.J;Zhou.A2001a

@article{Xu.J;Zhou.A2001a,
Abstract = {In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.},
Author = {Jinchao Xu and A. Zhou},
Title = {Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems},
Journal = {Advances in Comp. Math.},
Volume = {14},
Pages = {293--327},
Year = {2001},
Note = {},
}

Xu.J;Zhou.A2001

@article{Xu.J;Zhou.A2001,
Abstract = {A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.},
Author = {Jinchao Xu and A. Zhou},
Title = {A two-grid discretization scheme for eigenvalue problems},
Journal = {Mathematics of Computation},
Volume = {70},
Pages = {17--25},
Year = {2001},
Note = {},
}

Xu.J;Ying.L2001

@article{Xu.J;Ying.L2001,
Abstract = {An explicit upwind finite element method is given for the numerical computation to multi-dimensional scalar conservation laws. It is proved that this scheme is consistent to the equation and monotone, and the approximate solution satisfies discrete entropy inequality. To guarantee the limit of approximate solutions to be a measure valued solution, we prove an energy estimate. Then the Lp strong convergence of this scheme is proved.},
Author = {Jinchao Xu and L. Ying},
Title = {Convergence of an explicit upwind finite element method to multi-dimensional conservation laws},
Journal = {J. of Comp. Math.},
Volume = {19},
Pages = {87--100},
Year = {2001},
Note = {},
}

Wu.J;Srinivasan.V;Xu.J2001

@article{Wu.J;Srinivasan.V;Xu.J2001,
Abstract = {Numerical solutions to partial differential equations form the backbone of mathematical models that simulate the behavior of various electrochemical systems, specifically batteries and fuel cells. In this paper, we present a set of numerical algorithms that are applied to efficiently solve this system of equations. These fast algorithms are identified by fully understanding the physics of the problem and recognizing the strength of the coupling between the governing equations. We illustrate this coupling, specifically in the two potential equations, and demonstrate the need for their simultaneous solution using Newton method. We take a 2D thermal and electrochemical coupled Liion model and extend the oftused Band(J) subroutine by utilizing a Krylov iterative solver, GMRES, instead of the direct solver (Gauss elimination), to improve the solution efficiency of the large, nonsymmetric Jacobian system. In addition, we use a nonlinear GaussSeidel method to provide the initial guess for the Newton iteration, and precondition the GMRES solver with a Block GaussSeidel and Multigrid algorithm with a smoother based on the Tridiagonal Matrix Algorithm (TDMA). Every stage in this process has been seen to add to the efficiency of the resulting computer simulation with the final result being a substantial improvement in computation speed, namely simulating complete discharge of the cell in less than 10 mins for grid size of 45*32. },
Author = {J. Wu and V. Srinivasan and Jinchao Xu and C. Y. Wang},
Title = {Newton-Krylov-Multigrid method for battery simulation},
Journal = {J. of the Electrochemical Society},
Volume = {149},
Pages = {1342--1348},
Year = {2001},
Note = {},
}

Shu.S;Xiao.Y;Xu.J2001

@inproceedings{Shu.S;Xiao.Y;Xu.J2001,
Abstract = {},
Author = {S. Shu and Y. Xiao and Jinchao Xu and L. Zikatanov},
Title = {Algebraic multigrid methods for lattice block materials},
Journal = {},
Volume = {},
Pages = {287--306},
Year = {2001},
Note = {},
}

Shen.J;Wang.F;Xu.J2000

@article{Shen.J;Wang.F;Xu.J2000,
Abstract = {This paper concerns the iterative solution of the linear system arising from the Chebyshev collocation approximation of second-order elliptic equations and presents and optimal multigrid preconditioner based on alternating line Gauss-Seidel smoothers for the corresponding stiffness matrix of bilinear finite elements on the Chebyshev-Gauss-Lobatto grid.},
Author = {J. Shen and F. Wang and Jinchao Xu},
Title = {A finite element multigrid preconditioner for Chebyshev-collocation method},
Journal = {Applied Numerical Mathematics},
Volume = {33},
Pages = {471--477},
Year = {2000},
Note = {},
}

Xu.J;Zhou.A2000

@article{Xu.J;Zhou.A2000,
Abstract = {A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory.},
Author = {Jinchao Xu and A. Zhou},
Title = {Local and parallel finite element algorithms based on two-grid discretizations},
Journal = {Mathematics of Computation},
Volume = {69},
Pages = {881--909},
Year = {2000},
Note = {},
}

Xu.J;Zikatanov.L1999

@article{Xu.J;Zikatanov.L1999,
Abstract = {A simple technique is given in this paper on the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimensions by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element (EAFE) scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems},
Author = {Jinchao Xu and L. Zikatanov},
Title = {A Monotone Finite Element Scheme for Convection Diffusion Equations},
Journal = {Mathematics of Computation},
Volume = {68},
Pages = {1429--1446},
Year = {1999},
Note = {},
}

Wang.F;Xu.J1999

@article{Wang.F;Xu.J1999,
Abstract = {This paper is on a block iterative algorithm for convection-dominated equations. The algorithm uses cross-wind thin blocks in a block Gauss-Seidel method. The relaxation sweep is carried out successively along the downwind direction and exact solvers are used for the block systems. The method is efficient for convection dominated problems, discretized by monotone finite element, finite difference schemes, such as edge average finite element method. An optimal ordering and partitioning algorithm, Tarjan's algorithm is used to partition the nodes into cross-wind blocks and order the blocks in downwind direction. The convergence of this block iterative method is analysed and exponential convergence rates are proven for both one and two dimensional cases, on both structured and unstructured meshes.},
Author = {F. Wang and Jinchao Xu},
Title = {A cross-wind strip block iterative method for convection-dominated problems},
Journal = {SIAM Journal on Comput.},
Volume = {21},
Pages = {646--665},
Year = {1999},
Note = {},
}

Huang.Y;Xu.J1999

@inproceedings{Huang.Y;Xu.J1999,
Abstract = {In this paper some results are given on a new finite element method for solving elliptic problems with highly oscillatory coeficients. Local partial differential equations are solved to capture the oscillatory portions of the solution and a partition of unity is used to glue the local solutions together to obtain a global solution.},
Author = {Y. Huang and Jinchao Xu},
Title = {A partition-of-unity finite element method for elliptic problems with highly oscillating coefficients},
Journal = {},
Volume = {},
Pages = {},
Year = {1999},
Note = {},
}

Shen.L;Xu.J1998

@article{Shen.L;Xu.J1998,
Abstract = {This paper is on the study of a Schur complement operator arisen from the full discretization of the time-dependent Navier-Stokes equations and some inverse, stability and approximation properties are established. These properties indicate that the Schur complement operator behaves, in some sense, like a Laplacian operator discretized on a mesh of size comparable to the square root of temporal step-size, and they are instrumental for studying effective iterative methods for solving the underlying algebraic systems. As an application, they will be used to analyze a two-grid preconditioner for the Schur complement operator by the framework of {\it auxiliary space method} and the resulting analysis appears to be more transparent than what is available in the literature.},
Author = {L. Shen and J. Xu},
Title = {On A Schur Complement Operator Arisen from Navier-Stokes Equations and Its Preconditioning},
Journal = {Advances in Comp. Math. Proceedings of the Guangzhou International Symposium, edited by Zhongying Chen, Yuesheng Li, Charles a. Micchelli and Yuesheng Xu, Marcel Dekker, Inc.},
Volume = {},
Pages = {481-490},
Year = {1998},
Note = {},
}

Xu.J;Zou.J1998

@article{Xu.J;Zou.J1998,
Abstract = {The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuring--type methods and the Neumann--Neumann-type methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include local-global and global-local techniques. The analyses for both two- and three-dimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived.},
Author = {Xu, J. and Zou, J.},
Title = {Some nonoverlapping domain decomposition methods},
Journal = {SIAM Rev.},
Volume = {40},
Pages = {857--914},
Year = {1998},
Note = {},
}

Xu.J;Zhou.A1998

@inproceedings{Xu.J;Zhou.A1998,
Abstract = {This paper is devoted to the study of some local and parallel properties of finite elements for elliptic boundary value problems of second order. Several local and parallel algorithms are proposed and analyzed by means of two-grid discretiza- tions. The algorithms are motivated from the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One ma jor technical tool for the analysis is some sharp local a priori estimates for finite element solutions on general shape-regular grids. This paper can be considered as a sequel of our earlier paper [12] on a similar topic. While in [12] we studied our methods for a rather general class of model partial di?erential equations, in this paper we focus our attention on a rather specific case, namely, Neumann boundary value problems on a smooth domain and as a result we obtain some better error estimates for higher order elements.},
Author = {Jinchao Xu and A. Zhou},
Title = {Some local and parallel properties of finite element discretizations},
Journal = {},
Volume = {},
Pages = {},
Year = {1998},
Note = {},
}

Cheng.H;Yeh.G;Xu.J1998

@article{Cheng.H;Yeh.G;Xu.J1998,
Abstract = {},
Author = {H. Cheng and G. Yeh and Jinchao Xu and J. Cheng and R. Carsel},
Title = {A study of incorporating the multigrid method into the three-dimensional finite element discretization: a modular setting of grid generation and application},
Journal = {International Journal for Numerical Methods in Engineering},
Volume = {41},
Pages = {499--526},
Year = {1998},
Note = {},
}

Chan.T;Xu.J;Zikatanov.L1998

@inproceedings{Chan.T;Xu.J;Zikatanov.L1998,
Abstract = {A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of space. Our algorithm seems to be able to solve, for example, the Poisson equation discretized on any shape-regular finite element grids with nearly optimal complexity.},
Author = {Tony F. Chan and Jinchao Xu and Ludmil Zikatanov},
Title = {An agglomeration multigrid method for unstructured grids},
Journal = {},
Volume = {218},
Pages = {67--81},
Year = {1998},
Note = {},
}

Xu.J1997b

@inproceedings{Xu.J1997b,
Abstract = {An introduction is given in this paper to the basic idea and the convergence theory of multigrid methods. Brief discussions are first given to some basic properties of some elementary linear iterative methods such as Jacobi and Gauss-Seidel iterations and preconditioned conjugate gradient methods, and then more detailed discussions are given to a general framework of subspace correction method that can be applied to, among many other things, multigrid methods. A framework of auxiliary space method is also briefly presented for the construction of preconditioners. The multigrid method is introduced with a model elliptic boundary value problem of second order. Convergence estimates are obtained for basic multigrid methods such as backslash (\) cycle, V-cycle and W-cycle. Two different approaches are used in the convergence analysis. The first approach is the more traditional one that makes crucial use of elliptic regularity, while the second approach is based on the subspace correction framework that very weakly depends on the elliptic regularity. The first approach gives more precise estimates for simpler problems, while the second approach can be applied to more complex problems such as locally refined meshes and interface problems with large discontinuous jumps. As some more advanced topics, a general framework is briefly described on multigrid methods for nonnested multilevel subspaces and varying bilinear forms, and an optimal multigrid peconditioning technique is given for general unstructured grids using the auxiliary space framework. In addition to the aforementioned theoretical analysis, some discussions are also given to the implementation of some basic multigrid algorithms.},
Author = {Jinchao Xu},
Title = {An introduction to multigrid convergence theory},
Journal = {},
Volume = {},
Pages = {},
Year = {1997},
Note = {},
}

Xu.J1997a

@article{Xu.J1997a,
Abstract = {An introduction is given in this paper to the basic idea and the convergence theory of multilevel methods including overlapping domain decomposition methods and multigrid methods. Brief discussions are first given of some basic properties of some elementary linear iterative methods such as Jacobi and Gauss--Seidel iterations and preconditioned conjugate gradient methods, and then more detailed discussions are devoted to a general framework of subspace correction methods that can be applied to, among many other things, multilevel methods. A framework of auxiliary space methods is also briefly presented for the construction of preconditioners. The overlapping domain decomposition method and multigrid method are introduced with a model elliptic boundary value problem of second order. Convergence estimates are obtained for an overlapping domain decomposition method and especially for basic multigrid methods such as backslash (\) cycle, V-cycle and W-cycle. Two different approaches are used in the convergence analysis for multigrid methods. The first approach is the more traditional one that makes crucial use of elliptic regularity, while the second approach is based on the subspace correction framework that very weakly depends on the elliptic regularity. The first approach gives more precise estimates for simpler problems, while the second approach can be applied to more complex problems such as locally refined meshes and interface problems with large discontinuous jumps. As some more advanced topics, a general framework is briefly described on multigrid methods for nonnested multilevel subspaces and varying bilinear forms, and an optimal multigrid preconditioning technique is given for general unstructured grids using the auxiliary space framework. For nonsymmetric and/or indefinite problems, some two grid techniques are discussed. In addition to the aforementioned theoretical analysis, some discussions are also devoted to the implementation of some basic multigrid algorithms.},
Author = {Jinchao Xu},
Title = {An introduction to multilevel methods},
Journal = {},
Volume = {},
Pages = {},
Year = {1997},
Note = {published by Oxford University Press, New York},
}

Xu.J1997

@inproceedings{Xu.J1997,
Abstract = {In this paper, we discuss a special finite element scheme for convection diffusion problems and a domain decomposition method designed for such a discretization. The special discretization technique, known as edge-average finite element (EAFE in short) method, is obtained by properly averaging the PDE coefficients on element edges. When it is applied to convection diffusion equations in conservative form (in any spatial dimensions), the resulting finite element stiffness matrix is an M-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. Thus the method is particularly effective for convection dominated problems and furthermore the resulting algebraic system can be solved with a special domain decomposition method, known as cross-wind strip (CWS in short) method, with subdomains consisting of thin strips cross convection direction. Some numerical examples will be given.},
Author = {Jinchao Xu},
Title = {The {EAFE} scheme and {CWS} method for Convection Dominated Problems},
Journal = {},
Volume = {},
Pages = {},
Year = {1997},
Note = {},
}

Xu.J;Zhang.S1997

@article{Xu.J;Zhang.S1997,
Abstract = {This paper is concerned with the Poincar\'e-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincar\'e-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincar\'e-Steklov operator with respect to either a line (edge) or a star shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk or the web as a union of radii of a disk and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.},
Author = {Jinchao Xu and S. Zhang},
Title = {Preconditioning the Steklov-Poincaré operator by using Green's function},
Journal = {Mathematics of Computation},
Volume = {66},
Pages = {125--138},
Year = {1997},
Note = {},
}

Xu.J1996a

@article{Xu.J1996a,
Abstract = {An abstract framework ofauxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a furthernested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.},
Author = {Jinchao Xu},
Title = {The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids},
Journal = {Computing},
Volume = {56},
Pages = {215--235},
Year = {1996},
Note = {},
}

Xu.J1996b

@article{Xu.J1996b,
Abstract = {A number of finite element discretization techniques based on two (or more) subspaces for noninear elliptic partial differential equations is presented. Convergence estimates are derived to justify the efficiency of these algorithms. With the new proposed techniques, solving a large class of nonlinear elliptic boundary value problems will not much more expensive than the slution of one linearized equations. Similar techniques are also used to solve nonsymmetric and/or indefinite linear systems by solving symmetric positive definite systems. For the analysis of these two- or multi-grid methods, optimal $L^p$ error estimates are also obtained for the classic finite element discretizations.},
Author = {Jinchao Xu},
Title = {Two-grid discretization techniques for linear and nonlinear {P}{D}{E}s},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {33},
Pages = {1759-1777},
Year = {1996},
Note = {},
}

Xu.J1996c

@inproceedings{Xu.J1996c,
Abstract = {Various multigrid and domain decomposition methods are reviewed in this paper mainly for a model elliptic boundary value problem of second order. With some general frameworks such as space decomposition and subspace correction, main ideas are explained for most methods in review and relevant references are cited for further details. Classic multigrid schemes, hierarchical basis and BPX preconditioners are presented with sketch of proofs for main convergence results for nested quasiuniform meshes and briefly discussed for nonnested and locally refined meshes. Domain decomposition methods studied include nonoverlapping methods (such as substructuring methods) based on local Dirichlet problems and the methods (such as Neumann-Neumann and balancing domain decomposition methods) based on local Neumann problems; overlapping methods such as Schwarz methods are also discussed and their connection with some nonoverlapping methods are observed. Close relationship between numerous variations of these methods are investigated by means of some new techniques proposed in this paper.},
Author = {Jinchao Xu},
Title = {Iterative methods by multigrid and domain decomposition},
Journal = {},
Volume = {},
Pages = {229-253},
Year = {1996},
Note = {},
}

Bank.R;Xu.J1996

@article{Bank.R;Xu.J1996,
Abstract = {We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order $J^2$, where J is the number of hierarchical basis levels.},
Author = {R. E. Bank and Jinchao Xu},
Title = {An algorithm for coarsening unstructured meshes},
Journal = {Numerische Mathematik},
Volume = {73},
Pages = {1--36},
Year = {1996},
Note = {},
}

Marion.M;Xu.J1995

@article{Marion.M;Xu.J1995,
Abstract = {A new nonlinear Galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations. The new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size $H$ and one fine grid with grid size $h\ll H$. Nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time. With linear finite element discretizations, it is proved that the difference between the new nonlinear Galerkin solution and the standard Galerkin solution in $H^1(\Omega)$ norm is of the order of $H^3$.},
Author = {M. Marion and Jinchao Xu},
Title = {Error estimates on a new nonlinear Galerkin method based on two-grid finite elements},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {32},
Pages = {1170--1184},
Year = {1995},
Note = {},
}

Bank.R;Xu.J1995

@incollection{Bank.R;Xu.J1995,
Abstract = {This paper is concerned with the application of the hierarchical basis method to completely unstructured grids in two dimensional space. A two level method is presented and analyzed. Generalization to multilevel case is also discussed with numerical examples demonstrating the efficiency of the algorithm.},
Author = {R. E. Bank and Jinchao Xu},
Title = {A hierarchical basis multigrid method for unstructured grids},
Journal = {},
Volume = {49},
Pages = {1--13},
Year = {1995},
Note = {},
}

Xu.J1995

@incollection{Xu.J1995,
Abstract = {},
Author = {J. Xu},
Title = {Multigrid and domain decomposition methods},
Journal = {},
Volume = {},
Pages = {},
Year = {1995},
Note = {},
}

Xu.J1994a

@inproceedings{Xu.J1994a,
Abstract = {In this paper, we shall discuss a class of finite element discretization techniques based on two finite element space on one coarse and one fine grid. On the fine spac, only symmetric positive definite equations need to be solved for a nonsymmetric or indefinite linear equation and only one linear equation need to be solved for a nonlineaer equation. It is shown that the coarse grid can be extremely coarse to still achieve the optimal approximation on the fnie space for these algorithm. A special nonlineaer Galerkin method based on two-grid finite elements is also discussed.},
Author = {Jinchao Xu},
Title = {Some two-grid finite element methods},
Journal = {},
Volume = {157},
Pages = {79--87},
Year = {1994},
Note = {},
}

Xu.J1994

@article{Xu.J1994,
Abstract = {A new finite element discretization technique based on two (coarse and fine) subspaces is presented for a semilinear elliptic boundary value problem. The solution of a nonlinear system on the fine space is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse space and a linear system on the fine space. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation. As a result, the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation.},
Author = {Jinchao Xu},
Title = {A novel two-grid method for semilinear elliptic equations},
Journal = {SIAM Journal on Scientific Computing},
Volume = {15},
Pages = {231--237},
Year = {1994},
Note = {},
}

Xu.J;Qin.J1994

@article{Xu.J;Qin.J1994,
Abstract = {This paper is devoted to a class of multilevel preconditioners developed in [Math Comp., 55 (1990), pp. 1-22] by Bramble, Pasciak, and Xu and in the Ph.D. thesis of Xu at Cornell University, Ithaca, NY, 1989. A simple proof is given for optimal estimation of the conditioning. A derivation of matrix representations and a description of efficient implementation techniques are also given. Some modified preconditioners are proposed combining the hierarchical basis method. Numerical examples are also given that compare various preconditioners discussed in this paper.},
Author = {Jinchao Xu and J. Qin},
Title = {Some remarks on a multigrid preconditioner},
Journal = {SIAM Journal on Scientific Computing},
Volume = {15},
Pages = {172--184},
Year = {1994},
Note = {},
}

Bank.R;Xu.J1994

@inproceedings{Bank.R;Xu.J1994,
Abstract = {A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be effectively applied to problems discretized on completelyunstructured grids. Numerical experiments demonstrating the efficiency of the method are also reported.},
Author = {R. E. Bank and Jinchao Xu},
Title = {The hierarchical basis multigrid method and incomplete {LU} decomposition},
Journal = {},
Volume = {180},
Pages = {163--173},
Year = {1994},
Note = {},
}

Xu.J1992a

@article{Xu.J1992a,
Abstract = {The main purpose of this paper is to give a rather systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works on iterative methods, we shall present a unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss-Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods, multilevel nodal basis preconditioners and hierarchical basis methods. By using the notions of space decomposition and subspace correction, all these algorithms are classified into two groups, namely parallel subspace correction (PSC) and successive subspace correction (SSC) methods. These two types of algorithms are similar in nature to the familiar Jacobi and Gauss-Seidel methods respectively. A feature of our framework is that a quite general abstract convergence theory can be established. In order to apply the abstract theory to a particular problem, we need only to specify a decomposition of the underlying space and the corresponding subspace solvers. For example, subspaces arising from the domain decomposition method are associated with subdomains whereas with the multigrid method subspaces are provided by multiple ``coarser'' grids. By estimating only two parameters, optimal convergence estimations for a given algorithm can be obtained as a direct consequence of the abstract theory.},
Author = {Jinchao Xu},
Title = {Iterative methods by space decomposition and subspace correction},
Journal = {SIAM Review},
Volume = {34},
Pages = {581--613},
Year = {1992},
Note = {},
}

Xu.J1992b

@inproceedings{Xu.J1992b,
Abstract = {This paper is devoted a class of iterative methods for solving nonsymmetric or indefinite problems that are dominated by some SPD (symmetric positive definite) problems. The algorithm is based on a direct solver for the original equation restricted on a small subspace and a given iterative method for the SPD equation. It is shown that any convergent iterative method for the SPD problem will give rise to an algorithm that converges with a comparable rate if the small subspace is properly chosen. Furthermore a number of preconditioners that can be used with GMRES type methods are also obtained.},
Author = {Jinchao Xu},
Title = {Iterative methods by {SPD} and small subspace solvers for nonsymmetric or indefinite problems},
Journal = {},
Volume = {},
Pages = {106--118},
Year = {1992},
Note = {},
}

Xu.J1992

@article{Xu.J1992,
Abstract = {A new technique is proposed to solve NSPD (nonsymmetric or indefinite) problems that are "compact" perturbations of some SPD (symmetric positive definite) problems. In the new algorithm, a direct method is first used to solve the original equation restricted on a coarser space (that has a considerably smaller dimension), then a SPD equation for the residue is solve by using one or a few iterations of a given iterative algorithm. It is shown that for any convergent iterative method for the SPD problem, the new algorithm always converges with essentially the same rate if the coarse space is properly chosen. In applications, for multiplicative domain decomposition methods, the algorithm consists of solving the original NSPD problem on the coarse mesh and solving SPD equations on all subdomains; for multigrid methods, except when the correction on the coaarset mesh is first performed for the original NSPD equation, all smoothings are carried out for SPD equations on all other levels. It is shown that most of these algorithms converge uniformly provided that the coarsest meshsize is reasonaly small (but independent of the fine meshsize). Discussions are also devoted to some popular algorithms such as SOR method.},
Author = {Jinchao Xu},
Title = {A New class of iterative methods for nonselfadjoint or indefinite problems},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {29},
Pages = {303--319},
Year = {1992},
Note = {},
}

Shann.W;Xu.J1992

@article{Shann.W;Xu.J1992,
Abstract = {Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given},
Author = {W. Shann and Jinchao Xu},
Title = {Galerkin-wavelet method for two-point boundary value problems},
Journal = {Numerische Mathematik},
Volume = {63},
Pages = {123--144},
Year = {1992},
Note = {},
}

Cai.X;Xu.J1992

@article{Cai.X;Xu.J1992,
Abstract = {A preconditioned technique is proposed for nonsymmetric or indefinite linear systems of equations. The main idea in our theory, roughly speaking, is first to use some "coarser mesh" space to correct the nonpositive portion of the eigenvalues of the underlying operator and then switch to use a symmetric positive definite preconditioner. The generality of our theory allows us to apply any known preconditioners that were orginally designed for symmetric positive definite problems to nonsymmetric or indefinite problems, without losing the optimality that the original one has. Some numerical experiments based on GMRES are reported.},
Author = {X.--C. Cai and Jinchao Xu},
Title = {A preconditioned {GMRES} method for nonsymmetric or indefinite problems},
Journal = {Mathematics of Computation},
Volume = {59},
Pages = {311--319},
Year = {1992},
Note = {},
}

Bramble.J;Pasciak.J;Xu.J1992

@inproceedings{Bramble.J;Pasciak.J;Xu.J1992,
Abstract = {In this note, we consider multilevel preconditioning of the reduced boundary systems which arise in non-overlapping domain decomposition methods. It will be shown that the resulting preconditioned systems have condition numbers which are bounded in the case of multilevel spaces on the whole domain and grow at most proportional to the number of levels in the case of multilevel boundary spaces without multilevel extensions into the interior.},
Author = {J. H. Bramble and J. E. Pasciak and Jinchao Xu},
Title = {A multilevel preconditioner for domain decomposition boundary systems},
Journal = {},
Volume = {},
Pages = {},
Year = {1992},
Note = {},
}

Bramble.J;Pasciak.J;Wang.J1991a

@article{Bramble.J;Pasciak.J;Wang.J1991a,
Abstract = {In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces. The simplest algorithm of this sort has an error-reducing operator which is the product of orthogonal projections onto the complement of the subspaces. New norm-reduction estimates for these iterative techniques will be presented in an abstract setting. Applications are given for overlapping Schwarz algorithms with many subregions for finite element approximation of second-order elliptic problems.},
Author = {J. H. Bramble and J. E. Pasciak and J. Wang and Jinchao Xu},
Title = {Convergence estimates for product iterative methods with applications to domain decomposition},
Journal = {Mathematics of Computation},
Volume = {57},
Pages = {1--21},
Year = {1991},
Note = {},
}

Bramble.J;Pasciak.J;Wang.J1991

@article{Bramble.J;Pasciak.J;Wang.J1991,
Abstract = {A new technique for proving rate of convergence estimates of multi-grid algorithms for symmetric positive definite problems will be given in this paper. The standard multigrid theory requires a "regularity and approximation" assumption. In contrast, the new theory requires only an easily verified approximation assumption. This leads to convergence results for multigrid refinement applications, problems with irregular coefficients, and problems whose coefficients have large jumps. In addition, the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications.},
Author = {J. H. Bramble and J. E. Pasciak and J. Wang and J. Xu},
Title = {Convergence estimates for multigrid algorithms without regularity assumptions},
Journal = {Math. Comp.},
Volume = {57},
Pages = {23-45},
Year = {1991},
Note = {},
}

Xu.J1991

@article{Xu.J1991,
Abstract = {Counter examples are given to show that some results concerning a weighted $L^2$ projection presented earlier by Bramble and the author are sharp, i.e., that certain error and stability estimates are impossible in some cases.},
Author = {Jinchao Xu},
Title = {Counter examples concerning a weighted ${L}^{2}$ projection},
Journal = {Mathematics of Computation},
Volume = {57},
Pages = {563--568},
Year = {1991},
Note = {},
}

Bramble.J;Xu.J1991

@article{Bramble.J;Xu.J1991,
Abstract = {This paper is devoted to the error estimates for some weighted $L^2$ projections. Nearly optimal estimates are obtained. These estimates can be applied to the analysis of the usual multigrid method, multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jum discontinuities.},
Author = {J. H. Bramble and Jinchao Xu},
Title = {Some estimates for a weighted ${L}^2$ projection},
Journal = {Mathematics of Computation},
Volume = {56},
Pages = {463--476},
Year = {1991},
Note = {},
}

Bramble.J;Pasciak.J;Xu.J1991

@article{Bramble.J;Pasciak.J;Xu.J1991,
Abstract = {We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the $\mathscr{V}$ cycle, $\mathscr{W}$ cycle and a $\mathscr {V}$ cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way. We apply this theory to various numerical approximations of second-order elliptic boundary value problems. In our first example, we consider certain finite difference multigrid algorithms. In the second example, we consider a finite element multigrid algorithm with nested spaces, which however uses a prolongation operator that does not coincide with the natural subspace imbedding. The third example gives a multigrid algorithm derived from a loosely coupled sequence of approximation grids. Such a loosely coupled grid structure results from the most natural standard finite element application on a domain with curved boundary. The fourth example develops and analyzes a multigrid algorithm for a mixed finite element method using the so-called Raviart-Thomas elements.},
Author = {J. H. Bramble and J. E. Pasciak and Jinchao Xu},
Title = {The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms},
Journal = {Mathematics of Computation},
Volume = {56},
Pages = {1--34},
Year = {1991},
Note = {},
}

Bramble.J;Pasciak.J;Xu.J1990

@article{Bramble.J;Pasciak.J;Xu.J1990,
Abstract = {In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other well-known multilevel approaches.},
Author = {Bramble, James H. and Pasciak, Joseph E. and Xu, Jinchao},
Title = {Parallel Multilevel Preconditioners},
Journal = {Mathematics of Computation},
Volume = {55},
Pages = {1-22},
Year = {1990},
Note = {},
}

Xu.J1990

@inproceedings{Xu.J1990,
Abstract = {},
Author = {Jinchao Xu},
Title = {Convergence estimates for some multigrid algorithms},
Journal = {},
Volume = {},
Pages = {174--187},
Year = {1990},
Note = {},
}

Bramble.J;Pasciak.J;Xu.J1990a

@inproceedings{Bramble.J;Pasciak.J;Xu.J1990a,
Abstract = {},
Author = {J. H. Bramble and J. E. Pasciak and Jinchao Xu},
Title = {Parallel multilevel preconditioners},
Journal = {},
Volume = {},
Pages = {341--357},
Year = {1990},
Note = {},
}

Thomee.V;Zhang.N;Xu.J1989

@article{Thomee.V;Zhang.N;Xu.J1989,
Abstract = {Some recent results concerning maximum-norm superconvergence of the gradient in piecewise linear finite-element approximations of an elliptic problem are carried over to a parabolic problem. Both the standard semidiscrete in space Galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations, and the Crank-Nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes.},
Author = {V. Thomee and N. Zhang and Jinchao Xu},
Title = {Superconvergence of gradient in piecewise linear finite element approximation to a parabolic problem},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {26},
Pages = {553--573},
Year = {1989},
Note = {},
}

Xu.J1989

@phdthesis{Xu.J1989,
Abstract = {This thesis is devoted to the analysis for algorithms of multilevel types. This includes the ordinary multigrid methods as well as the preconditioners constructed from some multilevel spaces. An attempt is made to develop a unified theory. Comprehensive convergence estimates are established for various algorithms and a number of new algorithms are proposed and analyzed; Our focus is on the application of the multilevel algorithms to the finite element discretizations for second order elliptic boundary--value problems. In the preliminary chapters, results of the finite element spaces, new and old alike, are presented to meet the need of the analysis in multilevel algorithms. Some standard techniques in the ordinary multigrid method literature are summarized in an abstract form. For the ordinary multigrid algorithms, we study both the symmetric positive problems and nonsymmetric or indefinite problems. For the symmetric positive definite problems, we establish a general framework that would unify and generalize most of the existing theory in this direction. This theory allow the multilevel spaces not to be nested and the bilinear forms on different levels not to be strictly related. Convergence of the algorithms with a given number of smoothing steps are estabished under rather weak assumptions. A typical example of the sonvergence result is that the variable \vv--cycle algorithm gives rise to an optimal preconditioner even though it (as well as any other standard algorthms) may not have the usual reduction property. Applications of this theory include a number of new algorithms for problems defined on curved--boundary--domains. With some different approach, we also study the multigrid algortihm for interface problems with large jump coefficients, problems with arbitrary locally quaiuniform grids, interface problems with refined meshes and nonconforming finite element methods etc. For the nonsymmetric problems, we first give a simplified proof and improvement of a known result for the so-called nonsymmetric scheme. We then present a complete analysis for the so-called symmetric scheme. Various convergence results are established. The proofs depend on several technical lemmas. Finally we introduce a new multigrid preconditioner for symmetric problems. in contrast to the so-called heirachical preconditioner, it does not depend on the number of dimensions and is fully parallelizable. It also has the feature, as does the heirachical one, the relative condition number is good with the minimal elliptic regularity in the underlying problem.},
Author = {Jinchao Xu},
Title = {Theory of Multilevel Methods},
Journal = {},
Volume = {},
Pages = {},
Year = {1989},
Note = {},
}

Bramble.J;Xu.J1989

@article{Bramble.J;Xu.J1989,
Abstract = {A general simple post-processing technique is given in this paper. Under certain conditions, if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection Pu, then by this method, a better approximation for u can be easily obtained. Essentially, the leading term of the error between u and its original approximation can be computed. It is shown how the method may be applied to various mixed type finite-element methods.},
Author = {J. H. Bramble and Jinchao Xu},
Title = {A local post-processing technique for improving the accuracy in mixed finite element approximation},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {26},
Pages = {1267--1275},
Year = {1989},
Note = {},
}

Xu.J;Ying.L1988

@article{Xu.J;Ying.L1988,
Abstract = {},
Author = {J. Xu and L. Ying},
Title = {The analysis and improvement on infinite element algorithms},
Journal = {Chinise J. Numer. Math & Appl.},
Volume = {10},
Pages = {71-82},
Year = {1988},
Note = {},
}

Bramble.J;Pasciak.J;Xu.J1988

@article{Bramble.J;Pasciak.J;Xu.J1988,
Abstract = {We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable V-cycle and the V-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the V-cycle algorithm also cconverges (under appropriate assumptions on the coarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the V-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.},
Author = {J. H. Bramble and J. E. Pasciak and Jinchao Xu},
Title = {The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems},
Journal = {Mathematics of Computation},
Volume = {51},
Pages = {389--414},
Year = {1988},
Note = {},
}

Xu.J1985

@inproceedings{Xu.J1985,
Abstract = {This paper is concerned with the theoretical analysis and the practiccal algorithms for the infinite element method which was developed in recent years. With the aid of the inequalities related to the Sobolev space given in the paper, the error estimates in the norms of ____ (with weight), and ___ and ___, the estimates like superconvergence and so on are comprehensively investigated for the method. Many of the conclusions obtained are optimal or nearly optimal. Also, some relative algorithms are analyzed and improved and the results are suported by numerical experiments},
Author = {Jinchao Xu},
Title = {The error analysis and the improved algorithms for the infinite element method},
Journal = {},
Volume = {},
Pages = {326--331},
Year = {1985},
Note = {},
}

Lin.Q;Xu.J1985

@article{Lin.Q;Xu.J1985,
Abstract = {},
Author = {Q. Lin and Jinchao Xu},
Title = {Linear finite elements with high accuracy},
Journal = {J. Comp. Math.},
Volume = {3},
Pages = {115--133},
Year = {1985},
Note = {},
}

Xu.J1984

@article{Xu.J1984,
Abstract = {},
Author = {Jinchao Xu},
Title = {Some inequalities in Sobolev space and finite and infinite elements on polygonal domains},
Journal = {M. A. thesis, under Longan Ying, Peking University},
Volume = {},
Pages = {},
Year = {1984},
Note = {},
}

Xu.J1982

@article{Xu.J1982,
Abstract = {},
Author = {Jinchao Xu},
Title = {Error estimates of the finite element method for the 2nd order elliptic equation with discontinuous coefficient},
Journal = {J. Xiangtan Univ},
Volume = {},
Pages = {},
Year = {1982},
Note = {},
}