Subspace Correction Methods (MG, DDM)

• Randomized and Fault-Tolerant Method of Subspace Corrections

X. Hu, J. Xu, L. Zikatanov
Research in the Mathematical Sciences ,  6 (3),  29  (2019)

• Block triangular preconditioning for stochastic Galerkin method

D. Wang, B. Zheng, L. Chen, G. Lin, J. Xu
Journal of Computational and Applied Mathematics   (2022)

• An Error-Resilient Redundant Subspace Correction Method

T. Cui, J. Xu, and C. Zhang
Comput. Visual Sci. ,  18(2),  65--77  (2017)

• A Robust Multigrid Method for Discontinuous Galerkin Discretizations of Stokes and Linear Elasticity Equations

Q. Hong, J. Kraus, J. Xu, and L. Zikatanov
Numerische Mathematik ,  132(1),  23-49  (2016)

• Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

B. Zheng, L. Chen, X. Hu, L. Chen, R. Nochetto, and J. Xu
J. Sci. Comput. ,  69(1),  201--226  (2016)

• A Multigrid Solver based on Distributive Smoother and Residual Overweighting for Oseen Problems

L. Chen, X. Hu, M. Wang and J. Xu
Numer. Math. Theor. Meth. Appl.,  8(2),  237-252  (2015)

• Optimal Solvers for Fourth-Order PDEs Discretized on Unstructured Grids

S. Zhang and J. Xu
SIAM J. Numer. Anal.,  52(1),  282-307  (2014)

• Numerical Study of Geometric Multigrid on CPU--GPU Heterogenous Computers

C. Feng, S. Shu, J. Xu and C. Zhang
Adv. Appl. Math. Mech.,  6,  1-23  (2014)

• A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem

B. Ayuso de Dios, F. Brezzi, L. D. Marini, J. Xu and L. Zikatanov
J. Sci. Comp.,  58(3),  517-547  (2014)

• The Single-Grid Multilevel Method and Its Applications

J. Xu
Inverse Probl. Imaging,  7(3),  987-1005  (2013)

• Comparative Convergence Analysis of Nonlinear AMLI-cycle Multigrid

X. Hu and P. S. Vassilevski and J. Xu
SIAM J. Numer. Anal.,  51(2),  1349-1369  (2013)

• Convergence of Numerical Analysis of V-cycle Multigrid Methods for Anisotropic Elliptic Equations

Y. Wu, L. Chen, X. Xie and J. Xu
IMA J. Numer. Anal.,  32(4),  1329-1347  (2012)

• Optimal Multilevel Methods for Graded Bisection Grids

L. Chen, R. H. Nochetto, and J. Xu
Numer. Math.,  120(1),  1-34  (2012)

• Optimal multilevel methods for $H({\rm grad})$, $H({\rm curl})$, and $H({\rm div})$ systems on graded and unstructured grids

J. Xu and L. Chen and R. H. Nochetto
in "Multiscale, Nonlinear and Adaptive Approximation", Springer, 599-659, (2009)

• New estimates for the rate of convergence of the method of subspace corrections

D. Cho and J. Xu and L. Zikatanov
Numer. Math. Theory Methods Appl.  1  44--56  (2008)

• A sharp convergence estimate for the method of subspace corrections for singular systems of equations

Y.-J. Lee and J. Wu and J. Xu and L. Zikatanov
Math. Comp.  77  831--850  (2008)

• Convergence analysis on iterative methods for semidefinite systems

J. Wu and Y.-J. Lee and J. Xu and L. Zikatanov
J. Comput. Math.  26  797--815  (2008)

• Robust subspace correction methods for nearly singular systems

Y.-J. Lee and J. Wu and J. Xu and L. Zikatanov
Math. Models Methods Appl. Sci.  17  1937--1963  (2007)

• On the convergence of iterative methods for semidefinite linear systems

Y.-J. Lee and J. Wu and J. Xu and L. Zikatanov
SIAM J. Matrix Anal. Appl.  28  634--641  (2006)

• Successive subspace correction method for singular system of equations

Y.-J. Lee and J. Xu and L. Zikatanov
Proceedings of the 14th International Conference on Domain Decomposition Method, 315--321  (2003)

• Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities

X.-C. Tai and B. Heimsund and J. Xu
Proceedings of the 13rd International Conference on Domain Decomposition Method, 127--138  (2002)

• The method of alternating projections and the method of subspace corrections in Hilbert space

J. Xu and L. Zikatanov
J. Amer. Math. Soc.  15  573--597  (2002)

• Global and uniform convergence of subspace correction methods for some convex optimization problems

X.-C. Tai and J. Xu
Math. Comp.  71  105--124  (2001)

• The method of subspace corrections

J. Xu
J. Comput. Appl. Math.  128  335--362  (2000)

• Subspace correction methods for convex optimization problems

X.-C. Tai and J. Xu
    130--139 (electronic)  (1999)

• Some nonoverlapping domain decomposition methods

J. Xu and J. Zou
SIAM Rev.  40  857--914  (1998)

• Preconditioning the Poincare-Steklov operator by using Green's function

J. Xu and S. Zhang
Math. Comp.  66  125--138  (1997)

• An Introdution to Multilevel Method

J. Xu
"Wavelets, Multilevel Methods and PDEs", (editors: Mark Ainsworth, Jeremy Levesley, Will Light, and Marco Marletta) Oxford University Press, New York      (1997)

• An Introdution to Multigrid Convergence Theory

J. Xu
Department of Mathematics, University of California, Los Angeles      (1996)

• A hierarchical basis multigrid method for unstructured grids

R. E. Bank and J. Xu
Fast Solvers for Flow Problems  49  1--13  (1995)

• Some remarks on a multigrid preconditioner

J. Xu and J. Qin
SIAM J. Sci. Comput.  15  172--184  (1994)

• The hierarchical basis multigrid method and incomplete LU decomposition

R. E. Bank and J. Xu
Contemporary mathematics, 180  163--173  (1994)

• Iterative methods by space decomposition and subspace correction

J. Xu
SIAM Rev.  34  581--613  (1992)

• Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems

J. Xu
Proceedings of the 5th International Conference on Domain Decomposition Method, 106--118  (1992)

• A preconditioned GMRES method for nonsymmetric or indefinite problems

J. Xu and X.-C. Cai
Math. Comp.  59  311--319  (1992)

• Convergence estimates for multigrid algorithms without regularity assumptions

J. H. Bramble and J. E. Pasciak and J. P. Wang and J. Xu
Math. Comp.  57  23--45  (1991)

• Convergence estimates for product iterative methods with applications to domain decomposition

J. H. Bramble and J. E. Pasciak and J. P. Wang and J. Xu
Math. Comp.  57  1--21  (1991)

• The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms

J. H. Bramble and J. E. Pasciak and J. Xu
Math. Comp.  56  1--34  (1991)

• Convergence estimates for some multigrid algorithms

J. Xu
Proceedings of the 3rd International Conference on Domain Decomposition Method, 174--187  (1990)

• Parallel multilevel preconditioners

J. H. Bramble and J. E. Pasciak and J. Xu
Math. Comp.   174--187  (1990)

• The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems

J. H. Bramble and J. E. Pasciak and J. Xu
Math. Comp.  51  389--414  (1988)

Auxiliary Space Preconditioning Methods

• A nearly optimal multigrid method for general unstructured grids

L. Gradsedyck, L. Wang and J. Xu
Numerische Mathematik ,  134(3),  637-666  (2016)

• Multilevel Preconditioners for Reaction-Diffusion Problems with Discontinuous Coefficients

T. Kolev, J. Xu, and Y. Zhu
J. Sci. Comput. ,  67(1),  324--350  (2016)

• Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids

L. Chen, M. Holst, J. Xu and Y. Zhu
Comput. Vis. Sci.,  15(5),  271-289  (2012)

• Fast Poisson-Based Solvers for Linear and Nonlinear PDEs

J. Xu
Proceedings of the International Congress of mathematicians 2010 (ICM 2010) (2010)

• Auxiliary Space Preconditoners for Mixed Finite Element Methods

R. S. Tuminaro and J. Xu and Y. Zhu
Domian Decomposition Methods in Science and Engineering XVII. 70, 99-109 (2009)

• Optimal algorithms for discretized partial differential equations

J. Xu
in "ICIAM 07 -- 6th International Congress on Industrial and Applied Mathematics", Eur. Math. Soc., Zurich, 409-444 (2009)

• Auxiliary space preconditioning for edge elements

R. Hiptmair and J. Xu
IEEE Trans. Magnetics  44  938--941  (2008)

• Nodal auxiliary space preconditioning in ${\bf H}({\bf curl})$ and ${\bf H}({\rm div})$ spaces

R. Hiptmair and J. Xu
SIAM J. Numer. Anal.  45  2483--2509 (electronic)  (2007)

• The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

J. Xu
Computing  56  215--235  (1996)

Algebraic Multigrid Methods

• Algebraic Multigrid Methods

J. Xu, L. Zikatanov
Acta Numerica ,  591--721  (2017)

• A Two-grid SA-AMG Convergence Bound that Improves When Increasing the Polynomial Degree

X. Hu, P.S. Vassilevski, and J. Xu
Numer. Linear Algebra Appl. ,  23(4),  746--771  (2016)

• A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

J. C. Urschel, X. Hu, J. Xu, and L. Zikatanov
J. Comp. Math.,  33,  209-226  (2015)

• A Parallel Auxiliary Grid Algebraic Multigrid Method for Graphic Processing Units

L. Wang, X. Hu, J. Cohen and J. Xu
SIAM J. Sci. Comput.,  35(3),  C263-C283  (2013)

• An algebraic multigrid method for higher-order finite element discretizations

S. Shu and D. Sun and J. Xu
Computing  77  347--377  (2006)

• An algebraic multigrid method for finite element systems on criss-cross grids

S. Shu and J. Xu and Y. Yang and H. Yu
Adv. Comput. Math.  25  287--304  (2006)

• Extending the applicability of multigrid methods

J. Brannick and M. Brezina and R. Falgout and T. Manteuffel and S. McCormick and J. Ruge and B. Sheehan and J. Xu and L. Zikatanov
J. Phys.: Conf. Ser.  46  443--452  (2006)

• Uniformly convergent multigrid methods for convection-diffusion problems without any constraint on coarse grids

H. Kim and J. Xu and L. Zikatanov
Adv. Comput. Math.  20  385--399  (2004)

• On an energy minimizing basis for algebraic multigrid methods

J. Xu and L. Zikatanov
Comput. Vis. Sci.  7  121--127  (2004)

• A multigrid method based on graph matching for convection-diffusion equations

H. Kim and J. Xu and L. Zikatanov
Numer. Linear Algebra Appl.  10  181--195  (2003)

• Algebraic multigrid method on lattice block materials

S. Shu and J. Xu and Y. Xiao and L. Zikatanov
Recent Progress in Computational and Applied PDEs, 289 -307, (2002)

• A crosswind block iterative method for convection-dominated problems

F. Wang and J. Xu
SIAM J. Sci. Comput.  21  620--645 (electronic)  (1999)

• An agglomeration multigrid method for unstructured grids

T. F. Chan and J. Xu and L. Zikatanov
  218  67--81  (1998)

Two-grid Methods

• Two-grid Methods for Time-harmonic Maxwell Equations,

L. Zhong, S. Shu, J. Wang, and J. Xu
Numer. Linear Algebra Appl.,  20(1),  93-111  (2013)

• Analysis of a two-level method for anisotropic diffusion equations on aligned and nonaligned grids

G. Yu, J. Xu and L. Zikatanov
Numer. Linear Algebra Appl. ,  20(5),  832-851  (2013)

• Numerical Solution to A Mixed Navier-Stokes/Darcy Model by the Two-grid Approach

M. Cai and M. Mu and J. Xu
SIAM J. Numer. Anal.  47  3325--3338  (2009)

• Local and parallel finite element algorithms for the {S}tokes problem

Y. He and J. Xu and A. Zhou and J. Li
Numer. Math.  109  415--434  (2008)

• A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow

M. Mu and J. Xu
SIAM J. Numer. Anal.  45  1801--1813  (2007)

• Local and parallel finite element algorithms for the Navier-Stokes problem

Y. He and J. Xu and A. Zhou
J. Comput. Math.  24  227--238  (2006)

• A two-grid discretization method for decoupling systems of partial differential equations

J. Jin and S. Shu and J. Xu
Math. Comp.  75  1617--1626 (electronic)  (2006)

• Local and parallel finite element algorithms for eigenvalue problems

J. Xu and A. Zhou
Acta Math. Appl. Sin. Engl. Ser.  18  185--200  (2002)

• Some multiscale methods for partial differential equations

J. Xu and A. Zhou
Contemporary Mathematics, 306  1--27  (2002)  

• Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems

J. Xu and A. Zhou
Adv. Comput. Math.  14  293--327  (2001)

• Local and parallel finite element algorithms based on two-grid discretizations

J. Xu and A. Zhou
Math. Comp.  69  881--909  (2000)

• A two-grid discretization scheme for eigenvalue problems

J. Xu and A. Zhou
Math. Comp.  70  17--25  (1999)

• Some local and parallel properties of finite element discretizations

J. Xu and A. Zhou
Proceedings of the 11st International Conference on Domain Decomposition Method, 140--147 (electronic)  (1999)

• Two-grid discretization techniques for linear and nonlinear PDEs

J. Xu
SIAM J. Numer. Anal.  33  1759--1777  (1996)

• Error estimates on a new nonlinear Galerkin method based on two-grid finite elements

M. Marion and J. Xu
SIAM J. Numer. Anal.  32  1170--1184  (1995)

• A novel two-grid method for semilinear elliptic equations

J. Xu
SIAM J. Sci. Comput.  15  231--237  (1994)

• Some two-grid finite element methods

J. Xu
Proceedings of the 6th International Conference on Domain Decomposition Method, 157  79--87  (1994)

• A New class of iterative methods for nonselfadjoint or indefinite problems

J. Xu
SIAM Journal on Numerical Analysis  29  303--319  (1992)

Finite Element Methods (Finite Volume Methods, Infinite Element Methods)

• Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

Q. Hong, J. Xu
Journal of Computational Mathematics, Vol.39, No.2, 283-310 ,  (2021)

• An Extended Galerkin Analysis for Elliptic Problems

Q. Hong, S Wu, J. Xu
Science China Mathematics, 64 ,  (2021)

• Finite Neuron Method and Convergence Analysis

J. Xu
Commun. Comput. Phys.Vol. 28, No. 5, pp. 1707-1745 ,  (2020)

• An Abstract Stabilization Method with Applications to Nonlinear Incompressible Elasticity

Q. Hong, C. Liu, J. Xu
Mathematica Numerical Sinica,, 42 (3), 298-209 ,  (2020)

• An Extended Galerkin Analysis for Linear Elasticity with Strongly Symmetric Stress Tensor

Q Hong, J Hu, L Ma, J Xu
arXiv ,  2002.11664 ,    (2020)

• High-order extended finite element methods for solving interface problems

Y Xiao, J Xu, F Wang
Computer Methods in Applied Mechanics and Engineering ,  364,  112964  (2020)

• A mixed discontinuous Galerkin method for linear elasticity with strongly imposed symmetry

F Wang, S Wu, J Xu
Journal of Scientific Computing ,  83 (1),  1-17  (2020)

• Simplex-Averaged Finite Element Methods for (grad), (curl), and (div) Convection-Diffusion Problems

S Wu, J Xu
SIAM Journal on Numerical Analysis ,  58 (1),  884-906  (2020)

• Generalized Gaffney inequality and discrete compactness for discrete differential forms

J. He, K. Hu, J. Xu
Numerische Mathematik ,  143 (4),  781-795  (2019)

• New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers

S. Gong, S. Wu, J. Xu
Numerische Mathematik ,  141 (2),  569--604  (2019)

• A Unified Study of Continuous and Discontinuous Galerkin Methods

Q. Hong, F. Wang, S. Wu, and J. Xu
SCIENCE CHINA Mathematics ,  62 (1),  1--32  (2019)

• Nonconforming Finite Element Spaces for 2m-th Order Partial Differential Equations on R^n Simplicial Grids When m=n+1

S. Wu, and J. Xu
Mathematics of Computation 88 (316),  531--551  (2019)

• New hybridized mixed methods for linear elasticity and optimal multilevel solvers

S Gong, S Wu, J Xu
Numer. Math. ,  1--36  (2018)

• A Unified Study of Continuous and Discontinuous Galerkin Methods

Q. Hong, F. Wang, S. Wu, and J. Xu
SCIENCE CHINA Mathematics ,  1--32  (2018)

• Nonconforming Finite Element Spaces for 2m-th Order Partial Differential Equations on R^n Simplicial Grids When m=n+1

S. Wu, and J. Xu
Math. Comp. (2018)

• Interior Penalty Mixed Finite Element Methods of Any Order in Any Dimension for Linear Elasticity with Strongly Symmetric Stress Tensor

S. Wu, S. Gong, and J. Xu
Math. Models Methods Appl. Sci. ,  27,  2711--2743  (2017)

• Optimal Finite Element Methods for Interface Problems

J. Xu and S. Zhang
DD22 proceeding,  (2015)

• Lower Bounds of the Discretization Error for Piecewise Polynomials

Q. Lin, H. Xie and J. Xu
Math. Comp. ,  83,  1-13  (2014)

• Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

J. Hu and J. Xu
J. Sci. Comput.,  55(1),  125-148  (2013)

• Minimal Finite Element Spaces for 2m-th-order Partial Differential Equations in R^n

J. Xu and M. Wang
Math. Comp.,  82,  25-43  (2013)

• A Discontinuous Galerkin Method for the Fourth-order Curl Problem

Q. Hong, J. Hu, S. Shu and J. Xu
J. Comput. Math.,  30(6),  565-578  (2012)

• Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations

L. Zhong, L. Chen, S. Shu, G. Wittum and J. Xu
Math. Comp.,  81,  623-642  (2012)

• New mixed finite elements for plane elasticity and Stokes equations

X. Xie and J. Xu
China Math., Vol. 54, 7,1499-1519. (2011)

• A nonconforming finite element method for fourth order curl equations in $\Bbb{R}^{3}$

B. Zheng and Q. Hu and J. Xu
Math. Comp.  80  1871--1886  (2011)

• New mixed finite elements for plane elasticity and Stokes equations

X. Xie and J. Xu
Sci. China Math.,  54(7),  1499-1519  (2011)

• A Generalization of The Vertex-centered Finite Volume Scheme to Arbitrary High Order

A. Vogel and J. Xu and G. Wittum
Comput. Vis. Sci.  13  221--228  (2010)

• Analysis of linear and quadratic simplicial finite volume methods for elliptic equations

J. Xu and Q. Zou
Numer. Math.  111  469--492  (2009)

• Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell's equations

L. Zhong and S. Shu and G. Wittum and J. Xu
J. Comput. Math.  27  563--572  (2009)

• Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models

X. Xie and J. Xu and G. Xue
J. Comput. Math.  26  437--455  (2008)

• A new class of Zienkiewicz-type non-conforming element in any dimensions

M. Wang and Z.-c. Shi and J. Xu
Numer. Math.  106  335--347  (2007)

• Some $n$-rectangle nonconforming elements for fourth order elliptic equations

M. Wang and Z.-C. Shi and J. Xu
J. Comput. Math.  25  408--420  (2007)

• The finite element approximation of the nonlinear Poisson-Boltzmann equation

L. Chen and M. J. Holst and J. Xu
SIAM J. Numer. Anal.  45  2298--2320  (2007)

• Nonconforming tetrahedral finite elements for fourth order elliptic equations

M. Wang and J. Xu
Math. Comp.  76  1--18 (electronic)  (2006)

• The Morley element for fourth order elliptic equations in any dimensions

M. Wang and J. Xu
Numer. Math.  103  155--169  (2006)

• Modified Morley element method for a fourth elliptic singular perturbation problem

M. Wang and J. Xu and Y. Hu
J. Comp. Math  24  113-120  (2006)

• Partition of unity method on nonmatching grids for the Stokes problem

C. Bacuta and J. Chen and Y. Huang and J. Xu and L. Zikatanov
J. Numer. Math.  13  157--169  (2005)

• An optimal streamline diffusion finite element method for a singularly perturbed problem

L. Chen and J. Xu
  383  191--201  (2005)

• Partition of Unity Method for Stokes Problem on Nonmatching Grids

C. Bacuta and J. Xu
Proceedings of the 2003 Copper Mountain Conference on Multigrid Methods (2003)

• Regularity estimates for elliptic boundary value problems in Besov spaces

C. Bacuta and J. H. Bramble and J. Xu
Math. Comp.  72  1577--1595 (electronic)  (2003)

• Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains

C. Bacuta and J. H. Bramble and J. Xu
J. Numer. Math.  11  75--94  (2003)

• A conforming finite element method for overlapping and nonmatching grids

Y. Huang and J. Xu
Math. Comp.  72  1057--1066  (2003)

• Some observations on Babuska and Brezzi theories

J. Xu and L. Zikatanov
Numer. Math.  94  195--202  (2003)

• Convergence of an explicit upwind finite element method to multi-dimensional conservation laws

J. Xu and L. Ying
J. of Comp. Math.  19  87--100  (2001)

• A finite element multigrid preconditioner for Chebyshev-collocation method

J. Shen and F. Wang and J. Xu
Applied Numerical Mathematics  33  471--477  (2000)

• A partition-of-unity finite element method for elliptic problems with highly oscillating coefficients

Y. Huang and J. Xu
  99  27--30  (1999)

• A monotone finite element scheme for convection-diffusion equations

J. Xu and L. Zikatanov
Math. Comp.  68  1429--1446  (1999)

• On a Schur complement operator arisen from Navier-Stokes equations and its preconditioning

L. Shen and J. Xu
  202  481--490  (1999)

• The EAFE scheme and CWS method for Convection Dominated Problems

J. Xu
Proceedings of the 9th International Conference on Domain Decomposition Method, (1997)

• Galerkin-wavelet methods for two-point boundary value problems

J. Xu and W. C. Shann
Numer. Math.  63  123--144  (1992)

• Some estimates for a weighted $L^2$ projection

J. H. Bramble and J. Xu
Math. Comp.  56  463--476  (1991)

• Counterexamples concerning a weighted $L^2$ projection

J. Xu
Math. Comp.  57  563--568  (1991)

• Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem

V. Thom?e and J. Xu and N. Y. Zhang
SIAM J. Numer. Anal.  26  553--573  (1989)

• A local post-processing technique for improving the accuracy in mixed finite-element approximations

J. H. Bramble and J. Xu
SIAM J. Numer. Anal.  26  1267--1275  (1988)

• The analysis and improvement on infinite element algorithms

J. Xu and L. Ying
Chinese J. Numer. Math & Appl.  10  71-82  (1988)

• Analysis and improvement of algorithms for the infinite element method

J. Xu and L. A. Ying
Math. Numer. Sinica  8  164--174  (1986)

• Linear finite elements with high accuracy

Q. Lin and J. Xu
J. Comput. Math.  3  115--133  (1985)

• Estimate of the Convergence Rate of the Finite element Solutions to Elliptic Equation of Second Order with Discontinuous Coefficients

J. Xu and  Y. Long
Chinese J. Numer. Math. & Appl.  10, 71--82, (1988)

• Estimate of the Convergence Rate of the Finite element Solutions to Elliptic Equation of Second Order with Discontinuous Coefficients

J. Xu
Natural Science Journal of Xiangtan University  1  1-5  (1982)