Maximum principle for the finite element solution of time dependent anisotropic diffusion problems

TL;DR

This paper establishes conditions under which finite element solutions of time-dependent anisotropic diffusion problems preserve the maximum principle, linking mesh geometry, time step size, and matrix properties.

Contribution

It provides new mesh and time step criteria ensuring maximum principle preservation for finite element methods in anisotropic diffusion problems.

Findings

Maximum principle holds under non-obtuse mesh angles in the specified metric.

Lumped mass matrices relax the lower bound requirement on time step size.

Numerical results confirm theoretical conditions and findings.

Abstract

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the $\theta$-method in time for solving time dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are non-obtuse and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay-type conditions. Numerical results are presented to verify the theoretical findings.