Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form

TL;DR

This paper introduces fast auxiliary space preconditioners for linear elasticity problems using mixed finite element methods, achieving mesh-independent conditioning and demonstrating efficiency through numerical tests.

Contribution

It develops new preconditioners based on a stability analysis in mesh-dependent norms, including a stabilized low order method and superconvergence results.

Findings

Conditioning numbers are mesh- and Lamé-constant independent.

Preconditioners significantly improve solver efficiency.

Numerical examples confirm theoretical bounds.

Abstract

A block diagonal preconditioner with the minimal residual method and a block triangular preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods of linear elasticity. They are based on a new stability result of 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 the displacement it is spectral equivalent to 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 Lam\'e 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 of Hu-Zhang element are obtained.