Abstract

Discrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

中文翻译：

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分段常数和分段仿射向量和张量场的离散亥姆霍兹分解

摘要

离散亥姆霍兹分解将单纯网格上的分段多项式向量场分解为有限元函数的分段梯度和旋转。本文简要回顾了文献中已建立的结果，这些结果都仅限于分段常数的最低阶情况。它的主要贡献包括将这些分解推广到 3D 以及根据 Fortin-Soulie 函数对分段仿射向量场进行新颖的分解。虽然经典的最低阶分解包括一个一致部分和一个非一致部分，但分段仿射向量场的分解需要在两个部分中进行非一致富集。该演示涵盖了两个和三个空间维度，以及斯托克斯方程背景下的偏张量场的推广以及线性弹性和四阶问题的对称张量场。虽然证明集中于可收缩域，但也讨论了乘法连通域和具有非连通边界的域的概括。