Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes

TL;DR

This paper analyzes the stability conditions of explicit one-step methods for finite element approximations of linear diffusion equations on anisotropic meshes, revealing how mesh geometry and diffusion influence stability bounds.

Contribution

It provides a tight bound on the maximum stable time step for explicit schemes considering mesh and diffusion effects, including both full and lumped mass matrices.

Findings

Stability bound is tight within a factor of 2(d+1) for any mesh and diffusion matrix.

Mesh geometry relative to the diffusion matrix critically affects stability.

Numerical results confirm the theoretical stability bounds.

Abstract

We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.