the Institute for Theoretical and Engineering Science
Department of Mathematics

University of Houston

Scientific Computing Seminar

Professor Zhimin Zhang
Department of Mathematics
Wayne State University

High Accuracy Eigenvalue Approximations

Thursday, November 1, 2007
3:00 PM- 4:00 PM
Room 634 S&R1

Abstract: Finite element approximations for the eigenvalue problem of the Laplace operator is discussed. A gradient recovery scheme is proposed to enhance the finite element solutions of the eigenvalues. By reconstructing the numerical gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an a posteriori error estimator to guide an adaptive mesh refinement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes.

Additional computational cost for this post-processing technique is only $O(N)$ ($N$ is the total degrees of freedom), comparing with $O(N^2)$ cost for the original problem.

Theoretical results can be summarized in the following:

1) Eigenfunctions are sufficiently smooth. Under uniform meshes or the Delaunay triangulation with regular refinement, the enhanced eigenvalue approximations for the linear element converge at rate $O(h^4)$ ($h$ is the maximum mesh size), while the original approximations converge at rate $O(h^2)$.

2) Eigenfunctions have corner singularities. Under a commonly used adaptive mesh refinement strategy, the enhanced eigenvalue approximations converge at rate $O(N^{-k/2-\rho})$ for some $\rho>0$ ($k=1$ for the linear element and $k=2$ for the quadratic element), while the original approximations converges at rate $O(N^{-k/2})$.

All above theoretical results are numerically verified.

This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.