一般凸泛函的有限元极小值的凸包性质和最大值原理
Convex Hull Property and Maximum Principle for Finite Element Minimisers of General Convex Functionals
AI总结：
针对一般凸泛函的有限元极小值，在单纯非钝网格上建立了凸包性质，并推广到非线性问题如p-Laplacian和平均曲率问题，同时证明了严格锐角三角剖分上的强离散凸包性质。
摘要：
凸包性质是从标量函数到向量函数的最大值原理的自然推广。有限元近似的最大值原理通常对于保持相应物理模型的定性性质至关重要。在这项工作中，我们在单纯非钝网格上为$¶_1$协调有限元建立了一个凸包性质。证明不依赖于偏微分方程的线性结构，而是直接处理凸能量泛函的极小值的性质。因此，该结果适用于非常一般的非线性偏微分方程，包括例如$p$-Laplacian和平均曲率问题。在标量方程的情况下，所引入的技术可用于证明非线性问题的标准离散最大值原理。最后，我们证明了在严格锐角三角剖分上的强离散凸包性质。
英文摘要：
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.
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