Convex Hull Property and Maximum Principle for Finite Element Minimisers of General Convex Functionals

TL;DR

This paper establishes a convex hull property for finite element minimizers of convex functionals, ensuring qualitative property preservation for a wide class of nonlinear PDEs on specific mesh types.

Contribution

It introduces a convex hull property for  conforming finite elements on non-obtuse meshes, applicable to general nonlinear PDEs without relying on PDE linearity.

Findings

Proves convex hull property for finite element minimizers on non-obtuse meshes.

Extends maximum principle concepts to nonlinear PDEs like p-Laplacian.

Establishes strong discrete convex hull property on strictly acute triangulations.

Abstract

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 $\P_1$ 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 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.