Research Results
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, multigrid (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 BramblePasciakXu
preconditioner (a basic algorithm for solving elliptic
PDEs) and the HiptmairXu
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 XuZikatanov
identity, giving the optimal theory for these methods. Other
examples include:

The first uniform convergence theories for the multiplicative DD
and MG methods without using the elliptic regularity
assumption;

Convergence estimates for multigrid algorithms without
regularity assumptions, J. Bramble, J. Pasciak, J. Wang, and J.
Xu, Math. Comp., 57, 2345, 1991

Convergence estimates for product iterative methods
with applications to domain decomposition, J. Bramble, J.
Pasciak, J. Wang, and J. Xu, Math. Comp., 57, 121, 1991

The algorithms and theories 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, 22942312, 2003

Asymptotically exact a posteriori error estimators. II.
General unstructured grids, R. Bank, and J. Xu, SIAM J. Numer.
Anal., 41, 23132332, 2003

The twogrid discretization methods;

A novel twogrid method for semilinear elliptic equations
, J. Xu, SIAM J. Sci. Comput., 15, 231  237, 1994

Twogrid discretization techniques for linear and
nonlinear PDEs, J. Xu, SIAM J. Numer. Anal., 33, 17591777,
1996

The new algorithm and theory for nonNewtonian flows with high
Weissengberg numbers;

New formulations, positivity preserving discretizations
and stability analysis for nonNewtonian flow models, Y.J.
Lee, and J. Xu, Comput. Methods Appl. Mech. Engrg. (195)
11801206, 2006

Global existence, uniqueness and optimal solvers of
discretized viscoelastic flow models, Y.J. Lee, J. Xu
and C. Zhang, M3AS, 21(8), 17131732, 2001

The only known canonical construction of finite element family
for any order of elliptic PDEs in any spatial dimensions;

Minimal Finite Element Spaces for $2m$thorder Partial
Differential Equations in $R^n$, M. Wang, and J. Xu, Math.
Comp., 82, 24  43, 2013

Nonconforming Finite Element Spaces for $2m$th Order Partial
Differential Equations in $R^n$ When $m=n+1$, S. Wu, and J.
Xu, Math. Comp., accepted, 2018.

$\mathcal{P}^m$ Interior Penalty Nonconforming Finite Element Methods
for $2m$th Order PDEs in $R^n$, S. Wu, and J. Xu,
2017.

The monolithic discretization and robust solvers for
fluidstructure interaction (FSI);

Full Eulerian Finite Element Method of a Phase Field Model
for Fluidstructure Interaction Problem
, P. Sun, J. Xu, and L. Zhang, Comput. Fluids, 90, 1  8, 2014

Wellposedness and robust preconditioners for discretized
fluidstructure 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
interaction
, K. Yang, P. Sun, L. Wang, J. Xu, and L. Zhang,
Comput. Methods in Appl. Mech. Eng., 311, 788  814, 2016
Turek FluidStructure Interaction Benchmark

Moving mesh for real hydroelectric generator


The structurepreserving discretization and robust solvers for
magnetohydrodynamics (MHD);

Stable Finite Element Methods Preserving $\nabla \cdot
B = 0$ Exactly for MHD Models, K. Hu, Y.
Ma, and J. Xu, Numer. Math., 135(2), 371396, 2017

Robust Preconditioners for Incompressible MHD Models
, Y. Ma, K. Hu, X. Hu, and J. Xu, J. Comput. Phys., 316, 721  746, 2016
Liddriven cavity, Re=400, Rm=400: (left) stream line of the
velocity; (right) distribution of the total magnetic field
Liddriven cavity, Re=400, Rm=400: number of iterations


Analysis of the numerical schemes for phase field models and interface problems;

On the Stability and Accuracy of Partially and Fully
Implicit Schemes for Phase Field Modeling
, J. Xu, Y. Li, and S. Wu, arXiv:1604.05402, 2016

Multiphase AllenCahn and CahnHilliard models and their
discretizations with the effect of pairwise surface
tensions, S. Wu, and J. Xu, J. Comput. Phys., 343, 1032, 2017

Highorder Extended Finite Element Methods for Solving
Interface Problems, F. Wang, Y. Xiao, and J. Xu, arXiv:1604.06171, 2016

Algebraic multigrid methods

Deep learning

MgNet: A Unified Framework of Multigrid and Convolutional Neural Network, J. He and J. Xu, Sci. China Math., 62, 2019

ReLU Deep Neural Networks and Linear Finite Elements, J. He, L. Li, J. Xu and C. Zheng, J. Comput. Math., 2019

Make $\ell_1$ Regularization Effective in Training Sparse CNN, X. Jia, L. Zhao, L. Zhang, J. He and J. Xu, arXiv:1807.04222, 2018

On the Approximation Properties of Neural Networks, J. Siegel and J. Xu, arXiv:1904.0231, 2019

Extended Regularized Dual Averaging, J. Siegel and J. Xu, arXiv:1904.02316, 2019
