Maximum principle for the finite element solution of time-dependent anisotropic diffusion problems (Q2864599)

The authors study the preservation of the maximum principle for the combination of the linear finite element method in space and the \(\theta\)-method in time for solving time-dependent anisotropic diffusion problems. They obtain 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. In particular, the lower bound requirement can be removed when a lumped mass matrix is used. For two-dimensional cases, the mesh and time step conditions can be replaced by weaker Delaunay-type conditions. Numerical results verify the theoretical findings.