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Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor

Shuonan Wu

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

E-mail Address: [email protected]

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Shihua Gong

Beijing International Center for Mathematical Research, Peking University, Beijing 100871, P. R. China

E-mail Address: [email protected]

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, and

Jinchao Xu

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0218202517500567Cited by:17 (Source: Crossref)

Abstract

We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal $L^{2}$ -error estimate for displacement and optimal $H_{h} (div)$ error estimate for stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the method can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.

Communicated by F. Brezzi

Keywords:

AMSC: 65N30, 65N15, 74B05

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