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The convex hull property is the natural generalization of maximum principlesfrom scalar to vector valued functions. Maximum principles for finite elementapproximations are often crucial for the preservation of qualitative propertiesof the respective physical model. In this work we develop a convex hullproperty for $\P_1$ conforming finite elements on simplicial non-obtuse meshes.The proof does not resort to linear structures of partial differentialequations but directly addresses properties of the minimiser of a convex energyfunctional. Therefore, the result holds for very general nonlinear partialdifferential equations including e.g. the $p$-Laplacian and the mean curvatureproblem. In the case of scalar equations the introduce techniques can be usedto prove standard discrete maximum principles for nonlinear problems. Weconclude by proving a strong discrete convex hull property on strictly acutetriangulations.

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