A Study on Anisotropic Mesh Adaptation for Finite Element Approximation of Eigenvalue Problems with Anisotropic Diffusion Operators

Abstract

Anisotropic mesh adaptation is studied for the linear finite element solution of eigenvalue problems with anisotropic diffusion operators. The $\mathbb{M}$-uniform mesh approach is employed with which any nonuniform mesh is characterized mathematically as a uniform one in the metric specified by a metric tensor. Bounds on the error in the computed eigenvalues are established for quasi-$\mathbb{M}$-uniform meshes. Numerical examples arising from the Laplace--Beltrami operator on parameterized surfaces and nonlinear diffusion problems in image processing and radiation hydrodynamics are presented. Numerical results show that anisotropic adaptive meshes can lead to more accurate computed eigenvalues than uniform or isotropic adaptive meshes. They also confirm the second order convergence of the error that is predicted by the theoretical analysis. The effects of approximation of curved boundaries on the computation of eigenvalue problems is also studied in two dimensions. It is shown that the initial mesh used to define the geometry of the physical domain should contain at least $\sqrt{N}$ boundary points to keep the effects of boundary approximation at the level of the error of the finite element approximation, where $N$ is the number of the elements in the final adaptive mesh. Only about $\sqrt[3]{N}$ boundary points in the initial mesh are needed for boundary value problems. This implies that the computation of eigenvalue problems is more sensitive to the boundary approximation than that of boundary value problems.