Least-squares methods for linear elasticity

摘要

This paper develops least-squares methods for the solution of linear elastic problems in both two and three dimensions. Our main approach is defined by simply applying the L-2 norm least-squares principle to a stress-displacement system: the constitutive and the equilibrium equations. It is shown that the homogeneous least-squares functional is elliptic and continuous in the H(div; Omega)(d) x H-1(Omega)(d) norm. This immediately implies optimal error estimates for finite element subspaces of H(div; Omega)(d) x H-1(Omega)(d). It admits optimal multigrid solution methods as well if Raviart-Thomas finite element spaces are used to approximate the stress tensor. Our method does not degrade when the material properties approach the incompressible limit. Least-squares methods that impose boundary conditions weakly and use an inverse norm are also considered. Numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.