Stability of explicit Runge-Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes

Abstract

We study the stability of explicit Runge-Kutta 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.