Uniform Preconditioners for High Order Finite Element Approximations of Planar Linear Elasticity

TL;DR

This paper introduces a new preconditioner for high order finite element methods in 2D linear elasticity that maintains a bounded condition number regardless of mesh size, polynomial degree, or material ratio, with computational costs comparable to existing methods.

Contribution

A novel preconditioner for high order finite element elastic problems that achieves parameter-independent condition numbers with efficient computational cost.

Findings

Condition number bounded independently of p, h, and material ratio

Condition number roughly 6.0 on standard tests

Computational cost comparable to standard domain decomposition preconditioners

Abstract

A new preconditioner is developed for high order finite element approximation of linear elastic problems on triangular meshes in two dimensions. The new preconditioner results in a condition number that is bounded independently of the degree $p$, the mesh-size $h$ and the ratio $\lambda/\mu$. The resulting condition number is reduced to roughly $6.0$ for all values of the parameters and discretization parameters on standard test problems. Crucially, the overall cost of the new preconditioner is comparable to the cost of applying standard domain decomposition based preconditioners.