An Extended Galerkin Analysis for Linear Elasticity with Strongly Symmetric Stress Tensor
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This paper presents an extended Galerkin analysis for various Galerkin methods of the linear elasticity problem. The analysis is based on a unified Galerkin discretization formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor σ_h, displacement u_h inside each element and the modifications of these two variables σ_h and ǔ_h on elementary boundaries. Motivated by many relevant methods in literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problem and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to infinity, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problem. Two families of inf-sup conditions, one known as H^1-philic and another known as H(div)-phillic, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed formulations and also offer a new and unified analysis for many existing methods in literature as a by-product.
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