Research Interest

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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.

  • Multilevel Adaptive Finite Element Methods.
  • One of my current projects is to apply ideas from multigrid methodology to further improve the efficiency of existing adaptive methods. Existing AFEM algorithms start from a coarse grid and then repeatedly refine without returning to the coarse grid. This successive localized approach is reminiscent of cascade multigrid, which is usually not as efficient, as robust or as widely applicable as the the classic V-cycle or W-cycle multigrid methods. Motivated by our understanding of the different performance of classic multigrid versus cascade multigrid, we propose to develop V-cycle-multigrid-like AFEM to improve the the efficiency and robustness of the classic cascade-multigrid-like AFEM, and, as a result, to develop and analyze AFEM with error indicators and moving mesh methods with monitor functions in a common framework.

  • Nonlinear Approximation Theory.
  • Classic linear approximation theory uses Soblev smoothness to characterize the approximability of functions on quasi-uniform grids. The lack of Soblev smoothness of the solution will result a deteriorate of convergent rate if one insists on quasi-uniform grids. Adaptive finite element methods use Besov spaces which is much weaker requirement, to improve the convergence rate. Nonlinear approximation theory characterizes the Besov smoothness in terms of approximability. A new trend is to rely the adaptive finite element methods on the nonlinear approximation theory.
    Along this direction, I have done several work. One is to study best approximation of functions with anisotropic singularities. The upper bound and lower bound appeared in [8] using a new fractional norm of second derivatives of object function which could lead to a new class of function spaces. In [10, 11], a kind of Besov norm is used in the theoretical analysis of a 1-D convection-dominated problem and a uniform second order convergence result is obtained for functions with boundary layers. In those papers we also show that an unstable scheme on uniform grid can be made stable on a properly adapted grid and the accuracy of the adapted scheme may depend on (perhaps surprisingly) the uniformity on the grid where the solution is smooth [10, 11]).

  • Superconvergence.
  • Another related approximation problem is the superconvergence between solution and its interpolation. It can be used to construct efficient a posterior error estimator for adaptive finite element methods. In [12], we give a simple proof of classic superconvergence result in a symmetric mesh. Also in [4] we give a superconvergence result in 3-D for grid with small perturbation. In [9], we show in an optimal mesh, we may have an exact gradient recovery scheme which can be thought as a preliminary example on the nonlinear superconvergence theory on adaptive grids.

  • Mesh Generation and Improvement.
  • By the study of approximation theory, we give cretier of optimal mesh by the approximation error and developed a class of new methods for the mesh improvements. We proposed a new concept Optimal Delaunay Triangulation [9] and develop mesh adaptation schemes based on this concept [7, 2]. In this approach, we are able to combine local refinement and moving mesh methods to get a new and more efficient methods. A preliminary result is reported in [7, 2]. We note that in [1] another research group use this approach to successfully generate 3-D meshes which has better quality than other methods.

  • Multigrid on Adaptive Grids.
  • After we get the discrete equation, we need to solve a large algebraic system in the form Ax = b. In most cases the matrix obtained by finite element methods is sparse and thus iterative methods is better than direct methods. An most efficient iterative method is Mutligrid methods. Now I am working on the multigrid on adaptive refined grids and have obtained some preliminary results.

  • Convergence of Adaptive Finite Element Methods.
  • There are basically two questions on the AFEM. Does it converge? If yes, how good it is? Those two questions have confirmative answers for Poisson equation. Recently we extend the convergence and optimality of AFEM to mixed finite element methods [5] and a nonlinear Poisson-Boltzmann equation [6].

  • Applications.
  • Now I am developing and maintaining a software AFEM@matlab which is a MATLAB package of adaptive finite element methods for stationary and evolution partial differential equations in two spatial dimensions [13, 3]. This package contains robust, efficient, and easy-following codes for the main building blocks of AFEMs. It will (1) speed up program development; (2) facilitate comparisons of ideas and results; (3) improve academic publications. In spite of its brevity, the package is by no means a “toy” software. All the codes are written and optimized using MATLAB’s vectored addressing and built-in functions. Preliminary numerical tests show that our program can solve a middle size problem in seconds on a desktop PC.
    Another application project is related to math biology. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in [6]. This article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta-function sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.