Some Research Projects
in Computational Mathematics

Jinchao Xu

Center for Computational Mathematics and Applications

1  Introduction

Our research is mainly on the design, analysis, and applications of numerical methods for partial differential equations. We are interested in the study of basic discretization schemes (such as finite element), grid adaptation and advanced iterative methods (such as multigrid and domain decomposition). We are also interested in applying these techniques to numerical simulations for practical problems arising for example from complex fluids, electromagnetics, material sciences, and fuel cells.

research/StackedGrid.png research/mg.png

2  Multigrid and Subspace Correction Methods

Collaborators: James Brannick, Ludmil Zikatanov, and C-S. Zhang (PSU), L. Grasedyck (MPI, Leipzig), G. Wittum (Heidelberg)
We are interested in the following basic problem: how to solve Ax=b?
Example of Solvers: Our work addresses how multigrid methods can be made
research/jump3d.png     research/jump3dmesh.png
We study these issues using the framework of space decomposition and subspace corrections for general iterative methods (including multigrid and domain decomposition methods) [11,14]. Over the years, we have developed a number of basic algorithms such as BPX (Bramble-Pasciak-Xu [1]) preconditioner and more recently the Hiptmair-Xu preconditioner for H(curl) and H(div) systems [5]. We are making major efforts in developing algebraic multigrid methods (AMG) with various practical applications, as well as algorithms for singular, indefinite and nearly singular problems [12,7,6].
The multilevel and domain decomposition methods are the most efficient methods for solving the linear systems which come from the finite element discretization of partial differential equations. Unfortunately, the convergence of these methods will deteriorate rapidly when high contrast coefficients are present. Since 80's, there are many attempts to design uniformly convergent multilevel methods for such problems, but success has been achieved only in special cases (for the low dimension problems, or restrictions on the distributions of the subdomains, where the coefficients have jumps). It is known that in the worst case, the convergence rate of the multigrid methods is r @ 1-Ch, and the condition number of the preconditioned system is k(BA) @ Ch where h is the meshsize of the finest grid.
Recently, by observing that there are only a few small eigenvalues for the diagonal scaled discrete system, we were able to analyze the eigenvalue distributions of the BPX, multigrid and domain decomposition preconditioners in [13]. Our results show that the preconditioned system, has only a fixed number of small eigenvalues due the high contrast (jumps) in the PDE coefficients, and all the other eigenvalues are uniformly bounded with respect to the coefficients, and logarithmically with respect to the meshsize. An important result that we proved is that the asymptotic convergence rate of the corresponding preconditioned conjugate gradient (CG) algorithm is 1-[2/(C|logh|+1)].

3  Adaptive Mesh Refinement

Collaborators: L. Chen (UCI), P. Sun (UNLV), M. Holst (UCSD), R. Nochetto (UMD) and Ludmil Zikatanov, Y. Zhu, C-S. Zhang (PSU)
This project is focused on the construction of robust and accurate adaptive algorithms for mathematical models of physical phenomena that exhibit strong singularities.
research/adaptive.png
research/L.png
We give mathematically characterization of optimal or nearly optimal meshes for a general function which could be either isotropic or anisotropic. Roughly speaking, a nearly optimal mesh is a quasi-uniform triangulation under some new metric defined by the Hessian matrix of the object function. We also prove the error estimate is optimal for strictly convex (or concave) functions.
research/wellcontour.png research/wellmesh.png
Based on the interpolation error estimates, we introduce a new concept Optimal Delaunay Triangulation (ODT) and present practical algorithms to construct such nearly optimal meshes. By minimizing the interpolation error globally or locally, we obtain some new functionals for the moving mesh method and several new mesh smoothing schemes.
A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in the paper [3]. A regularized Poisson-Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the 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 a posteriori error estimate is shown to converge.
research/PBEmolecular.png
In a recent paper [4], optimal additive and multiplicative multilevel methods for solving H1 systems are designed and analyzed on adaptive grids obtained by bisections. We have designed a novel decomposition of spaces based on the geometric structure and a special smoothing for the newly added nodes, thus obtaining an optimal multigrid algorithm for locally refined grids.
In a forthcoming paper, we shall design and analyze adaptive multigrid methods for the H(curl) and H(div) systems in three dimensions. Our point of departure is the recent work on Hiptmair and Xu of additive preconditioners [5]. The analysis of multigrid for higher order Lagrange elements on the H1 system is an essential part of this study.
research/Decomposition123.png
research/Decomposition456.png

4  Auxiliary Space Maxwell Solver and H-X Preconditioner

Collaborators: R. Hiptmair (ETH), L. Zikatanov (Penn State), R. Tuminaro (Sandia National Lab), P. Vassilevski and T. Kolev (Lawrence Livermore National Lab), G. Wittum (University of Heidelberg), Y. Zhu (PSU)
Develop optimal and yet practical solvers for large scale (H(curl) and H(div)) systems of equations arising from, for example, electro-magnetics, MHD, subsurface flows and fuel cells. Comparing with the better-known Poisson equation (H(grad) system), the H(curl) and H(div) are much more difficult to solve because the operators curl and div have large kernels. Recently, Hiptmair and Xu developed an innovative multilevel method that have been proved optimal and also very easy to be applied in practical applications. It is now the most efficient and most robust method for these types of systems. But there are still a lot of open problems that need to be studied for this newly developed algorithm, especially, when various different systems are couple together. For example, in MHD application, Maxwell equations are coupled with Navier-Stokes equations.
research/maxwell1.png
Figure 1: 3d MHD Model Problem: our algorithm is 2 order of magnitude faster than the existing algorithms
LLNL's Hypre scalable solver library (http://www.llnl.gov/CASC/linear_solvers/) now includes our first provably scalable solver for the positive semi-definite form of Maxwell equations on general unstructured meshes.
research/maxwell3.png     research/maxwell4.png
Figure 2: Speedup by ASM solver

5  Multigrid Methods for Lattice QCD

Collaborators: V. Nistor, J. Brannick, and L. Zikatanov (PSU)
Lattice QCD calculations allow us to understand the results of particle and nuclear physics experiments in terms of QCD, the theory of quarks and gluons. The aim of the project is to develop numerical methods for lattice QCD calculations. A main focus is to develop multigrid solvers for the Dirac equation on a 4d hypercubic space-time lattice.
research/qcd1.png
This project is a close interdisciplianry collaboration between leading particle physicists at Boston University, computational mathematicians at CU Boulder, Lawrence Livermore National Laboratory, and here at PSU, and Computer Scientists from Argonne National Laboratory and University of Wuppertal, Germany.

6  Fuel Cell Modeling and Simulations

Collaborators: P. Sun (UNLV), C. Liu, C. Wang, G. Xue, L. Zikatanov (PSU)
research/fuelcell2.png
Fuel cell is a clean energy conversion device which has potential to substitute the internal combustion engine. A fuel cell has seven components: gas channel, gas diffusion layer (GDL), catalyst layer in both cathode and anode side, and proton exchange membrane. One key issue in the Proton Exchange Membrane (PEM) fuel cell, is the water management. Too much liquid water will block the open pores in the porous media of the Gas Diffusion Layer (GDL) and cover the surface of Catalyst Layer, too little water will dry out the membrane. In both cases, the fuel cell cannot operate.
Correct understanding of the liquid water transport in the fuel cell is essential to designing the fuel cell device. Mathematical modeling and numerical simulations are now being used to study this key issue.
A crucial part in the numerical simulations of fuel cell is the convergence of the iterative method for the solution of this nonlinear, coupled system. The convergence of such nonlinear iteration for steady state simulation is very slow if one uses commercial softwares like Star-CD or Fluent. We have recently developed our own Fuel Cell Simulator (FCS), which is robust with respect to all physical and chemical parameters and converges very fast. [10,9,2] See Figure 6.
research/fuelcell_o.png research/fuelcell_n.png
Figure 3: Convergence History of Fluent Simulation (Left) and our Fuel Cell Simulator (Right)

7  Non-Newtonian Fluids Simulation

Collaborators: Y-J. Lee (Rutgers Univ), X. Hu (Zhejiang Univ), J. Jia (Peking Univ), C-S. Zhang (PSU)
Fluids such as water and air are Newtonian. This means that a plot of shear stress versus shear rate at a given temperature is a straight line with a constant slope that is independent of the shear rate. Any fluid that does not obey the Newtonian relationship between the shear stress and shear rate is called non-Newtonian. The subject of theology is devoted to the study of the behavior of such fluids.
A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. As a result, non-Newtonian fluids may not have a well-defined viscosity. Although the concept of viscosity is commonly used to characterize a material, it can be inadequate to describe the mechanical behavior of a substance. They are best studied through several other rheological properties which relate the relations between the stress and strain tensors under many different flow conditions, such as oscillatory shear, or extensional flow which are measured using different devices or rheometers.
               
research/karman.png
Figure 4: von Karman vortex simulation
For higly elastic flows, characterized by the high Weissenberg number, the numerical simulations are notoriously difficult. For example, Oldroyd-B model in Riccati form
We have developed proper numerical schemes [8] that has been proven to be able to simulate such highly elastic flows. Much of our efforts have been devoted toward the unresolved challenge in non-Newtonian fluids simulations that has lasted for three decades.

8  Surface Water Waves

Collaborators: D. Henderson and M. Patterson (PSU)
The Pritchard Lab contains a wave basin with a segmented, programmable wave-maker system that is capable of generating both 2D and 3D water waves. These wave motions are typically modeled by the (inviscid) Euler equations assuming the flow to be irrotational. Yet, both viscous and rotational effects have been observed in many experiments. In particular, a remarkably stable 2D vortex has been observed in 2D and weakly 3D experiments. The vortex forms near the center of the basin, spanning its width,and then propagates slowly to the wave-maker where it is extinguished.
research/twophase7.png
Figure 5: The CCMA computing lab oversees the water wave tank.
We are currently using our grid adaptation and multigrid techniques to develop a numerical wave basin" based on the Navier-Stokes equations that will be used to predict wave motions as well as resulting vortical motions.

9  Black Hole Simulation

Collaborators: C. Sopuerta, P. Sun (UNLV), P. Laguna
Numerical evolutions of black holes have been improved slowly but steadily over the last few years and now first attempts are being made to extract physical information from these evolutions. The most notable one wants to predict the gravitational radiation emitted during black hole coalescence.
research/blackhole1.png research/blackhole2.pngresearch/blackhole3.png research/blackhole4.png
Figure 6: 3d adaptive mesh for black hole simulation
In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. Unphysical gravitational radiation present in the initial data will contribute to the gravitational waves computed in an evolution. Therefore an important question is how to control the gravitational wave content of initial-data sets, and how to specify astrophysically relevant initial data with the appropriate gravitational wave content, for e.g. two black holes orbiting each other.

References

[1]
J. H. Bramble, J. E. Pasciak, and J. Xu. Parallel multilevel preconditioners. Mathematics of Computation, 55(191):1-22, 1990.
[2]
J. Brannick, R. Falgout, J. Xu, G. Xue, and L. Zikatanov. Algebraic multigrid methods in direct methanol fuel cells. Preprint, 2007.
[3]
L. Chen, M. Holst, and J. Xu. The finite element approximation of the nonlinear poisson-boltzmann equation. SIAM Journal on Numerical Analysis, 45(6):2298-2320, 2007.
[4]
L. Chen, R. H. Nochetto, and J. Xu. Multilevel methods on graded bisection grids I: H1 system. Preprint, 2007.
[5]
R. Hiptmair and J. Xu. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. Research report no. 2006-09, ETH, Zurich, Switzerland, 2006.
[6]
Y. Lee, J. Wu, J. Xu, and L. Zikatanov. On the convergence of iterative methods for semidefinite linear systems. SIAM J. on Matrix Analysis, 2006. (to appear).
[7]
Y. Lee, J. Wu, J. Xu, and L. Zikatanov. Robust subspace correction methods for nearly singular systems. (in preparation), 2006.
[8]
Y.-J. Lee and J. Xu. New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg., 195(9-12):1180-1206, 2006.
[9]
P. Sun, G. Xue, C. Y. Wang, and J. Xu. A combined finite element-upwind finite volume method for three dimensional simulations of liquid feed direct methanol fuel cells. Preprint, 2007.
[10]
P. Sun, G. Xue, C. Y. Wang, and J. Xu. A domain decomposition method for two-phase mixture transport model in the cathode of a polymer electrolyte fuel cell. To be submitted, 2007.
[11]
J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Review, 34:581-613, 1992.
[12]
J. Xu. A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM Journal on Numerical Analysis, 29:303-319, 1992.
[13]
J. Xu and Y. Zhu. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. M3AS, 2007.
[14]
J. Xu and L. Zikatanov. The method of alternating projections and the method of subspace corrections in Hilbert space. Journal of The American Mathematical Society, 15:573-597, 2002.


File translated from TEX by TTH, version 3.79.
On 31 Dec 2007, 15:45.