Fast auxiliary space preconditioners for linear elasticity in mixed form.

A block-diagonal preconditioner with the minimal residual method and an approximate block-factorization preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods for linear elasticity. They are based on a new stability result for the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while for the displacement it is spectral equivalent to the Schur complement. A fast auxiliary space preconditioner based on the H-1-conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lame constant and the mesh size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results for the Hu-Zhang element are obtained.