Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems

A mesh condition is developed for linear finite element approximations of anisotropic diffusion–convection–reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and $$\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)$$O(‖b‖∞h+‖c‖∞h2)-nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where $$h$$h denotes the mesh size and $$\varvec{b}$$b and $$c$$c are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an $$\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)$$O(‖b‖∞h+‖c‖∞h2) perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.

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