Penalty-Free Any-Order Weak Galerkin FEMs for Linear Elasticity on Quadrilateral Meshes

This paper develops a family of new weak Galerkin (WG) finite element methods (FEMs) for solving linear elasticity in the primal formulation. For a convex quadrilateral mesh, degree k ≥ 0 vector-valued polynomials are used independently in element interiors and on edges for approximating the displacement. No penalty or stabilizer is needed for these new methods. The methods are free of Poisson-locking and have optimal order (k+1) convergence rates in displacement, stress, and dilation (divergence of displacement). Numerical experiments on popular test cases are presented to illustrate the theoretical estimates and demonstrate efficiency of these new solvers. Extension to cuboidal hexahedral meshes is briefly discussed.