Low-degree robust Hellinger-Reissner finite element schemes for planar linear elasticity with symmetric stress tensors
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In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconforming Helling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is span{([ 1 0; 0 0 ]),([ 0 1; 1 0 ]),([ 0 0; 0 1 ]),([ 0 x; x 0 ]),([ 0 y; y 0 ]),([ 0 x^2-y^2; x^2-y^2 0 ])}, the dimension of the total space is asymptotically 8 times of the number of vertices, and the supports of the basis functions are each a patch of an edge. The piecewise rigid body space is used for the displacement. Robust error estimations in 𝕃^2 and broken 𝐇( div) norms are presented. Secondly, we investigate the theoretical construction of schemes with lowest-degree polynomial shape function spaces. Specifically, a Hellinger-Reissner finite element scheme is constructed, with the local shape function space for the stress tensor being 5-dimensional which is of the lowest degree for the local approximation of 𝐇( div;𝕊), and the space for the displacement is piecewise constants. Robust error estimations in 𝕃^2 and broken 𝐇( div) norms are presented for regular solutions and data.
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