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Finite element approximations of symmetric tensors on simplicial grids in Rn: the high order case

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.