Numerical Solution of PDEs, Adaptive Finite Element Method, Grid Generation, Multigrid
2006 - 2007. Postdoc, Department of Mathematics, University of Maryland.
Advisor: Professor Ricardo Nochetto
2005 - 2006. Postdoc, Department of Mathematics, UCSD.
Advisor: Professor Michael Holst
My main research interest is the theoretical analysis and practical application of Adaptive Finite Element Methods (AFEMs). The numerical experiments using FEM need high accuracy to get reliable results. However high accuracy will increase the computation effort including the physical memory as well as cpu time. To speed up the simulation, AFEM is introduced to reduce the size of the computation while keeping optimal accuracy and thus now widely used in scientific computation. While these methods have been shown to be very successful, the theory ensuring the convergence of the algorithm and the advantages over non-adaptive methods is still under development. My main research goal is to to investigate a more complete integration of adaptive and multilevel algorithms, in terms of algorithm design, convergence and complexity theory, and application to important problems in science and engineering.
W. C. Lo, L. Chen, M. Wang, and Q. Nie. A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems. Accepted. Journal of Computational Physics, 2012.
Y. Wu, L. Chen, X. Xie, and J. Xu. Convergence Analysis of V-Cycle Multigrid Methods for Anisotropic Elliptic Equations. IMA Journal of Numerical Analysis 2012; doi: 10.1093/imanum/drr043.
L. Zhong, L. Chen, S. Shu, G. Wittum, and J. Xu. Quasi-optimal convergence of adaptive edge finite element methods for three dimensional indefinite time-harmonic Maxwell's equations. Mathematics of Computation. 81(278):623-642, 2012.
L. Chen, R.H. Nochetto and J. Xu. Optimal Multilevel Methods for Graded Bisection Grids. Numerische Mathematik. 120(1): 1-34, 2011.
L. Chen and M.J. Holst. Efficient Mesh Optimization Schemes based on Optimal Delaunay Triangulations. Computer Methods in Applied Mechanics and Engineering 200, 967–984, 2011.
H. Wei, L. Chen and Y. Huang. Superconvergence and Gradient Recovery of Linear Finite Elements for the Laplace-Beltrami Operator on General Surfaces. SIAM Journal on Numerical Analysis, 48(5):1920-1943, 2010.
L. Chen and C-S. Zhang. A coarsening algorithm on adaptive grids by newest vertex bisection and its applications. Journal of Computational Mathematics, 28(6):767-789, 2010.
L. Zhong, S. Shu, L. Chen, and J. Xu. Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numerical Linear Algebra with Applications. 17(2-3):415-432, 2010.
P. Sun, L. Chen, and J. Xu. Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems. 43(1), 24-43. Journal of Scientific Computing. 2010.
L. Chen. A New Class of High Order Finite Volume Methods for Second Order Elliptic equations. 47(6), 4021-4043. SIAM Journal on Numerical Analysis. 2010.
L. Chen and H. Li. Superconvergence of Gradient Recovery Schemes on graded meshes for corner singularities. Journal of Computational Mathematics. 28, 11-31, 2010.
L. Chen. On minimizing the linear interpolation error of convex quadratic functions and the optimal simplex. East Journal on Approximations, 14(3), 271--284, 2008.
L. Chen, M.J. Holst, and J. Xu. Convergence and optimality of adaptive mixed finite element methods. 78: 35--53, Mathematics of Computation, 2009.
L. Chen and J. Xu. Stability and accuracy of adapted finite element methods for singularly perturbed problems. Numerische Mathematik, 109(2): 167 - 191, 2008.
L. Chen, Y. Wang and J. Wu. Stability of a Streamline Diffusion Finite Element Methods for Turning Point Problems. Journal of Computational and Applied Mathematics, 220, 712 - 724, 2008.
L. Chen, M.J. Holst, and J. Xu. The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation. SIAM Journal on Numerical Analysis, 45(6): 2298-2320, 2007.
J. Jiang, S. Shu, Y. Huang, and L. Chen. A mesh adaptive method for two-dimensional three-tempeature heat conduction equations. Chinese Journal of Computational Physics, 24(1):19-28, 2007.
L. Chen, P. Sun, and J. Xu. Optimal anisotropic simplicial meshes for minimizing interpolation errors in $L^p$ norm. Mathematics of Computation, 76(257):179-204, 2007.
L. Chen. Superconvergence of tetrahedral linear finite elements. International Journal of Numerical Analysis and Modeling, 3(3):273-282, 2006.
L. Chen. New analysis of the sphere covering problems and optimal polytope approximation of convex bodies. Journal of Approximation Theory, 133(1):134-145, 2005.
L. Chen and J. Xu. Optimal Delaunay triangulations. Journal of Computational Mathematics, 22(2):299-308, 2004.
L. Chen. Deriving the X-Z Identity from Auxiliary Space Method. Domain Decomposition Methods in Science and Engineering XIX, 309-316, 2010.
L. Chen, R.H. Nochetto and C-S. Zhang. Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids. Domain Decomposition Methods in Science and Engineering XIX, 229-236, 2010.
L. Chen. Short bisection implementation in MATLAB. In Recent Advances in Computational Sciences -- Selected Papers from the International Workship on Computational Sciences and Its Education, 318 -- 332, 2008
L. Chen and J. Xu. An optimal streamline diffusion finite element method for a singularly perturbed problem. In AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, volume 383, pages 236-246, Hangzhou, 2005.
L. Chen, P. Sun, and J. Xu. Multilevel homotopic adaptive finite element methods for convection dominated problems. In The Proceedings for 15th Conferences for Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering 40, pages 459-468. Springer, 2004.
L. Chen. Mesh smoothing schemes based on optimal Delaunay triangulations. In 13th International Meshing Roundtable, pages 109-120, Williamsburg, VA, 2004. Sandia National Laboratories.
NSF Grant (PI) DMS-1115961 $179,987.00, Oct 2011 - Sept 2014
NSF Grant (PI) DMS-0811272 $149,999.00, Sept 2008 - Aug 2011