Full H(div)-approximation of linear elasticity on quadrilateral meshes based on ABF finite elements

For meshes of nondegenerate, convex quadrilaterals, we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity. The problem is posed in terms of the stress tensor, the displacement vector and the rotation scalar fields, with the symmetry of the stress tensor weakly imposed. The proposed spaces are based on the Arnold–Boffi–Falk (ABFk, k≥0) elements for the stress and piecewise polynomials for the displacement and the rotation. We prove that these finite elements provide full H(div)-approximation of the stress field, in the sense that it is approximated to order hk+1, where h is the mesh diameter, in the H(div)-norm. We show that displacement and rotation are also approximated to order hk+1 in the L2-norm. The convergence is optimal order for k≥1, while the lowest order case, index k=0, requires special treatment. The spaces also apply to both compressible and incompressible isotropic problems, i.e., the Poisson ratio may be one-half. The implementation as a hybrid method is discussed, and numerical results are given to illustrate the effectiveness of these finite elements.