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A hybridizable discontinuous Galerkin method for linear elasticity. (English) Zbl 1176.74196
Summary: This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of \(k+1\) for any \(k \geqslant 0\). The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of \(k+2\), whenever \(k \geqslant 2\). Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
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