Convex hull property and maximum principles for finite element minimizers of general convex functionals

Report

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are crucial for the preservation of qualitative properties of the physical model. In this work we develop a convex hull property for $P_{1}$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimizer of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $p$-Laplacian and the mean curvature problem. In the case of scalar equations the presented arguments can be used to prove standard discrete maximum principles for nonlinear problems.

Authors: Diening, L; Kreuzer, C; Schwarzacher, S

• Publisher: Numerische Mathematik
• Publication date: 2012-03-01