High Accuracy Eigenvalue Approximations
Dr. Zhimin Zhang
Department of Mathematics
Wayne State University
Abstract
Finite element approximations for the eigenvalue roblem 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 {\it 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.