Finite element approximations of symmetric tensors on simplicial grids in Rn: the high order case
For researchers in numerical analysis and computational mechanics, this provides a unified framework extending existing 2D and 3D methods to arbitrary dimensions, though it is an incremental extension of prior work by Hu and Zhang.
This paper constructs mixed finite element methods with symmetric stress approximations on simplicial grids in any space dimension, using lower-order polynomial spaces than previous methods. The discrete stress space is $H(\\d,\\Omega;\\mathbb{S})$---$P_k$ tensors and the displacement space is $L^2(\\Omega;\\mathbb{R}^n)$---$P_{k-1}$ vectors for $k\\geq n+1$.
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 $H(\d,Ω; \mathbb {S})$---$P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k\geq 2$.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 $H(\d,Ω; \mathbb {S})$---$P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k\geq 2$. 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 $H(\d,Ω;\mathbb {S})$---$P_k$ tensors, and the discrete displacement space here is the space of $L^2(Ω; \mathbb{R}^n)$---$P_{k-1}$ vectors for $k\geq n+1$. 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.