On the interplay between meshing and discretization in three-dimensional diffusion simulation

The maximum principle is the most important property of solutions to diffusion equations. Violation of the maximum principle by the applied discretization scheme is the cause for severe numerical instabilities: the emergence of negative concentrations and, in the nonlinear case, the deterioration of the convergence of the Newton iteration. We compare finite volumes (FV) and finite elements (FE) in three dimensions with respect to the constraints they impose on the mesh to achieve a discrete maximum principle. Distinctive mesh examples and simulations are presented to clarify the mutual relationship of the resulting constraints: Delaunay meshes guarantee a maximum principle for FV, while the recently introduced dihedral angle criterion is the natural constraint for FE. By constructing a mesh which fulfills the dihedral angle criterion but is not Delaunay we illustrate the different scope of both criteria. Due to the lack of meshing strategies tuned for the dihedral angle criterion we argue for the use of FV schemes in three-dimensional diffusion modeling.