Volume 33, Issue 3
Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case

J. Comp. Math., 33 (2015), pp. 283-296.

Published online: 2015-06

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• Abstract

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.

• Keywords

Mixed finite element, Symmetric finite element, First order system, Simplicial grid, Inf-sup condition.

65N30, 73C02.

hujun@math.pku.edu.cn (Jun Hu)

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@Article{JCM-33-283, author = {Hu , Jun}, title = {Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {3}, pages = {283--296}, abstract = {

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1412-m2014-0071}, url = {http://global-sci.org/intro/article_detail/jcm/9842.html} }
TY - JOUR T1 - Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case AU - Hu , Jun JO - Journal of Computational Mathematics VL - 3 SP - 283 EP - 296 PY - 2015 DA - 2015/06 SN - 33 DO - http://doi.org/10.4208/jcm.1412-m2014-0071 UR - https://global-sci.org/intro/article_detail/jcm/9842.html KW - Mixed finite element, Symmetric finite element, First order system, Simplicial grid, Inf-sup condition. AB -

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.

Jun Hu. (2019). Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case. Journal of Computational Mathematics. 33 (3). 283-296. doi:10.4208/jcm.1412-m2014-0071
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