On the Singular Neumann Problem in Linear Elasticity

TL;DR

This paper investigates well-posed formulations and preconditioning strategies for the singular Neumann problem in linear elasticity, proposing methods that improve accuracy and convergence, especially for mixed and displacement formulations.

Contribution

The paper introduces robust preconditioners for different formulations of the Neumann problem and proposes a modified conjugate gradient method respecting orthogonality constraints.

Findings

Preconditioners independent of discretization parameters.

Robust preconditioners for pure displacement and mixed formulations.

Modified conjugate gradient method achieves optimal error convergence.

Abstract

The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a non-singular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lam\'e constant are constructed for the pure displacement formulations, while a preconditioner that is robust in both Lam\'e constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. Based on this observation a modification to the conjugate gradient method is proposed that achieves optimal error convergence of the computed solution.