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.
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机译：凸包属性是从标量函数到矢量值函数的最大原理的自然概括。有限元逼近的最大原理对于保持物理模型的定性特性至关重要。在这项工作中，我们为简单非钝网格上的符合$ P_ {1} $的有限元开发了凸包属性。该证明不求助于偏微分方程的线性结构，而是直接解决了凸能量泛函的极小子的性质。因此，结果适用于非常普遍的非线性偏微分方程，例如包括$ p $ -Laplacian和平均曲率问题。在标量方程的情况下，提出的论点可用于证明非线性问题的标准离散最大原理。
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