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

Abstract

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ-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 nonobtuse 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.

A study on moving mesh finite element solution of the porous medium equation

“…4. How to suppress these oscillations using a monotone or structure-preserving scheme (e.g., see [8,41,42,44,46,48,57,58,59]) and to combine them with adaptive mesh movement for PME are worth future investigations.…”

A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations

“…We first consider the explicit method (29). Notice that u h ≤ 1 is equivalent to v h ≥ 0, where v h = 1 − u h .…”

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

“…Interesting features of PME also appear in APME such as finite propagation, free boundaries and waiting time phenomenon. Moreover, with the anisotropy of the porous media, satisfaction of maximum principle becomes more challenging and special mesh adaptation is needed, see [21,23,24] and the references therein. This paper serves as a starting effort about APME and its numerical solutions.…”