Convex hull property and maximum principle for finite element minimisers of general convex functionals
Convex hull property and maximum principle for finite element minimisers of general convex functionals
复制标题
一般凸函数有限元极小值的凸包性质和极大值原理
DOI：
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发表时间：
2013
影响因子：
2.1
通讯作者：
S. Schwarzacher
中科院分区：
文献类型：
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作者：
L. Diening;C. Kreuzer;S. Schwarzacher
凸包性质是最大值原理从标量函数到矢量值函数的自然推广。有限元近似的最大值原理对于保持各自物理模型的定性属性通常是至关重要的。在这项工作中，我们发展了单纯非钝化网格上$$mathbb{P}_1$$协调有限元的凸壳性质。证明不依赖于偏微分方程式的线性结构，而是直接涉及凸能量泛函的极小值的性质。因此，结果适用于非常一般的非线性偏微分方程组，包括$$p$$-拉普拉斯问题和平均曲率问题。在标量方程的情况下，引入的技巧可以用来证明非线性问题的标准离散极大值原理。最后，我们证明了严格锐化三角剖分的一个强离散凸包性质。
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 $$mathbb{P }_1$$ 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.