Research

Research Interests

My main research area is numerical analysis for PDEs by finite element methods such as nonconforming finite element methods for nonlinear problems, domain decomposition methods for linear and nonlinear problems, multilevel and multigrid methods for plate bending problem, moving mesh finite element methods for Navier-Stokes and variational inequality equations.

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The details are as follows:

1. Finite element approximation for some nonlinear equations.

(1) Maximum norm estimation for Navier-Stokes equation discretized by nonconforming finite elements.; (2) Nonconforming finite element methods with moving meshes for Navier-Stokes and variational inequality equations.
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2. Domain decomposition methods.

(1) Domain decomposition methods for some variational inequality equations.; (2) Domain decomposition methods for nonlinear equations.
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3. Multilevel and multigrid methods for PDEs.

(1) Construct efficient multigrid methods for nonconforming elements.; (2) Develop multilevel preconditioners for conforming and nonconforming plate elements.; (3) Design cascadic multigrid algorithms.
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Recent PapersĄĄ

Xuejun Xu, Shangyou Zhang, A new divergence-free interpolation operator with applications to the Darcy-Stokes-Brinkman equations, SIAM J. Sci. Comput., 2010. (to appear)

Xuejun Xu, Lizhen Qin, Spectral analysis of Dirichlet-Neumann operators and optimized Schwarz methods with Robin transmission conditions, SIAM J. Numer. Anal., 2010. (to appear)

Huangxin Chen, Xuejun Xu, Local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems, SIAM J. Numer. Anal., 2010.(to appear)

R. Talal, Xuejun Xu, A multilevel preconditioner for the mortar method for nonconforming P1 finite elementŁŹESIAM Mathematical Modelling and Numerical Analysis, 2009.

Lizhen Qin, Xuejun Xu, Optimized Schwarz Methods with Robin Transmission Conditions for Parabolic Problems, SIAM J. Sci. Comput., 2008 ĄĄ

T. Rahman, P. Bjorstad, Xuejun Xu, The Crouziex-Raviart FE on nonmatching grids with an approximate mortar condition, SIAM J. Numer. Anal., 2008.

Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang, The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes, Discrete Contin. Dyn. Syst. Ser. B, 2008.

Zhong-Ci Shi, Xuejun Xu, Yunqing Huang, Economical cascadic multigrid methods, Sci. in China, Ser. A., 2007. ĄĄ

Lizhen Qin, Xuejun Xu, On a parallel Robin-type nonoverlapping domain decomposition method, SIAM J. Numer. Anal., 2006.

Xuejun Xu, C.O. Chow, S.H. Lui, On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations, RAIRO M2AN, 2005.

Xuejun Xu, R.H.W. Hoppe, On the convergence of mortar edge element methods in R^3, SIAM J. Numer. Anal., 2005.

T. Rahman, Xuejun Xu, R.H.W. Hoppe, Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients, Numer. Math., 2005.

Xuejun Xu, S.H. Lui, Talal Rahman, A Two-level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation, IMA. J. Numer. Anal., 2004.

Xuejun Xu, J. Chen, Multigrid for the mortar element methods for parabolic problems, J. Comput. Math., 2003.

Xuejun Xu, Janka Ales, J.A. Desideri, Cascadic multigrid for the advection-diffusion equation, INRIA RR4574, 2002.

Xuejun Xu, Likang Li, Wenbin Chen, A multigrid method for the Mortar-type Morley element approximation of a plate bending problem, SIAM J. Numer. Anal., 2002.

Jinru Chen, Xuejun Xu, Uniform Convergence and Schwarz method for the Mortar projection nonconforming and mixed element methods for nonselfadjoint and indefinite problems, Comput. Methods Appl. Mech. Engrg., 2002.

Jinru Chen, Xuejun Xu, The Mortar element method for nonselfadjoint parabolic problem, Appl. Numer. Math. 2001.

Zhong-Ci Shi, Xuejun Xu, A new cascadic multigrid, Sciences in China (Series A), 2000.

Zhong-Ci Shi, Xuejun Xu, Cascadic multigrid for parabolic problem, J. Comput. Math., 2000.

Zhong-Ci Shi, Xuejun Xu, A V-cycle multigrid method for TRUNC plate element, Comput. Methods Appl. Mech. Engrg., 2000.

Xuejun Xu, On the accuracy of nonconforming quadrilateral Q1 element approximation of Navier-Stokes problem, SIAM J. Numer. Anal., 2000.

Xuejun Xu, Jinru Chen, Multigrid for the Mortar element method for P1 nonconforming element, Numer. Math., 2001.

Xuejun Xu, Multilevel methods for Wilson element approximation of elasticity problem, Comput. Methods Appl. Mech. Engrg., 174 (1999), 191-201.

Zhong-Ci Shi, Xuejun Xu, Cascadic multigrid method for elliptic problems, East-West J. Numer. Math., 7 (1999), 199-211.

Zhong-Ci Shi, Xuejun Xu, An analysis of V-cycle multigrid method for Wilson nonconforming element, Sciences in China (Series A), 29 (1999),880-891.

Xuejun Xu, Likang Li, A V-cycle multigrid method for the plate bending problem discreted by nonconforming finite elements, J. Comput. Math., 17 (1999), 533-544.

Zhong-Ci Shi, Xuejun Xu, A note on nonconforming multigrid method for nonsymmetric and indefinite problems, Math. Numer. Sinica, 21 (1999), 508-512. (in Chinese)

Xuejun Xu, Likang Li, A V-cycle multigrid method and addtive multilevel preconditioners for plate bending problem discretized by conforming finite elements, Appl. Math. and Comput., 93(1998), 253-258.

Xuejun Xu, Likang Li, Two-level Schwarz methods for Wilson element approximation of elaticitiy problem, Comput. Methods Appl. Mech. Engrg. , 142 (1997), 256-268.

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