Convex Hull Property and Maximum Principle for Finite Element Minimisers of General Convex Functionals
Provides a theoretical foundation for preserving qualitative properties in finite element approximations of nonlinear PDEs, extending maximum principles to vector-valued problems.
The paper develops a convex hull property for P1 conforming finite elements on simplicial non-obtuse meshes, applicable to general convex functionals. This yields discrete maximum principles for nonlinear PDEs like the p-Laplacian and mean curvature problem.
The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $¶_1$ 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 minimiser 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 introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.