My research ranges from
pure theoretical analysis, basic
algorithmic development to practical applications.
I mainly work on numerical methods for partial differential
equations (PDEs), such as finite element, multi-grid (MG) and
domain decomposition (DD) methods. Theoretical elegance and
practical usefulness can go together, and the design and analysis of
algorithms can be beautiful. Theory is the soul of what I do, and
practical needs are what motivate me. One thing that I very
much enjoy doing these days is to help people to speed up their
simulation codes from a few times to a few orders of magnigudes (for
very large scale problems). In all my work, I try to strike a
balance between rigor versus practicality.
Examples of my better known works include the Bramble-Pasciak-Xu
preconditioner (a basic algorithm for solving elliptic
PDEs) and the Hiptmair-Xu
preconditioner (an effective Maxwell solver which was
featured in a 2008 report by the U.S. Department of Energy as one of the top 10
breakthroughs in computational science in recent years).
I developed the framework and theory of the
Method of Subspace
Corrections that have been widely used in the
literature for the design and analysis of iterative methods and
later established the Xu-Zikatanov
identity, giving the optimal theory for these methods. Other
The first uniform convergence theories for the multiplicative DD
and MG methods without using the elliptic regularity
Convergence estimates for multigrid algorithms without
regularity assumptions, J. Bramble, J. Pasciak, J. Wang, and J.
Xu, Math. Comp., 57, 23--45, 1991
Convergence estimates for product iterative methods
with applications to domain decomposition, J. Bramble, J.
Pasciak, J. Wang, and J. Xu, Math. Comp., 57, 1--21, 1991
The ground-breaking algorithm and theory of optimal solver and
asymptotically exact a posteriori error estimators for PDEs
discretized on unstructured grids;
Asymptotically exact a posteriori error estimators. I.
Grids with superconvergence, R. Bank, and J. Xu, SIAM J. Numer.
Anal., 41, 2294--2312, 2003
Asymptotically exact a posteriori error estimators. II.
General unstructured grids, R. Bank, and J. Xu, SIAM J. Numer.
Anal., 41, 2313--2332, 2003
The two-grid discretization method that triggered a large number
of follow-up works;
A novel two-grid method for semilinear elliptic equations
, J. Xu, SIAM J. Sci. Comput., 15, 231 -- 237, 1994
Two-grid discretization techniques for linear and
nonlinear PDEs, J. Xu, SIAM J. Numer. Anal., 33, 1759--1777,
The new algorithm and theory for non-Newtonian flows with high
New formulations, positivity preserving discretizations
and stability analysis for non-Newtonian flow models, Y.-J.
Lee, and J. Xu, Comput. Methods Appl. Mech. Engrg. (195)
Global existence, uniqueness and optimal solvers of
discretized viscoelastic flow models, Y.-J. Lee, J. Xu
and C. Zhang, M3AS, 21(8), 1713--1732, 2001
The only known canonical construction of finite element family
for any order of elliptic PDEs in any spatial dimensions;
The monolithic discretization and robust solvers for
fluid-structure interaction (FSI);
Full Eulerian Finite Element Method of a Phase Field Model
for Fluid-structure Interaction Problem
, P. Sun, J. Xu, and L. Zhang, Comput. Fluids, 90, 1 -- 8, 2014
Well-posedness and robust preconditioners for discretized
fluid-structure interaction systems, J. Xu, and K. Yang,
Comput. Methods in Appl. Mech. Eng., 292, 69 -- 91, 2015
Modeling and simulation for fluid and rotating structure
, K. Yang, P. Sun, L. Wang, J. Xu, and L. Zhang,
Comput. Methods in Appl. Mech. Eng., 311, 788 -- 814, 2016
Turek Fluid-Structure Interaction Benchmark
Moving mesh for real hydroelectric generator
The structure-preserving discretization and robust solvers for
Stable Finite Element Methods Preserving $\nabla \cdot
B = 0$ Exactly for MHD Models, K. Hu, Y.
Ma, and J. Xu, Numer. Math., 2016
Robust Preconditioners for Incompressible MHD Models
, Y. Ma, K. Hu, X. Hu, and J. Xu, J. Comput. Phys., 316, 721 -- 746, 2016
Lid-driven cavity, Re=400, Rm=400: (left) stream line of the
velocity; (right) distribution of the total magnetic field
Lid-driven cavity, Re=400, Rm=400: number of iterations
Analysis of the numerical schemes for phase field models and
modeling of the multiphase problems;
Convex splitting schemes interpreted as fully implicit
schemes in disguise for phase field modeling, J. Xu, Y. Li, and S. Wu, arXiv:1604.05402, 2016
Multiphase Allen-Cahn and Cahn-Hilliard models and their
discretizations with the effect of pairwise surface
tensions, S. Wu, and J. Xu, J. Comput. Phys., 343, 10--32, 2017
High order finite elment methods for interface problems;
Algebraic Multigrid Methods and Deep Learning