A Note on the Effect of the Choice of Weak Form on GMRES Convergence for Incompressible Nonlinear Elasticity Problems

Abstract

The generalised minimal residual (GMRES) method is a common choice for solving the large non-symmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes' flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of Navier-Stokes' flow (where the matrix is block-symmetric/block-skew-symmetric); however there has been very little investigation into GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper we identify one feature of the problem formulation which has a huge impact on unpreconditioned GMRES-convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on total computation time.