Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

Abstract

Abstract This paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the L 2 {L^{2}} norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of H ⁢ ( div ; Ω ) {H(\mathrm{div};\Omega)} for the stress, of H 1 ⁢ ( Ω ) {H^{1}(\Omega)} for the displacement/velocity, and of L 2 ⁢ ( Ω ) {L^{2}(\Omega)} for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal L 2 {L^{2}} norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.