Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations

Abstract

This paper examines the performance of the Jacobi preconditioner when used with two Krylov subspace iterative methods. The number of iterations needed for convergence was shown to be different for drained, undrained and consolidation problems, even for similar condition number. The differences were due to differences in the eigenvalue distribution, which cannot be completely described by the condition number alone. For drained problems involving large stiffness ratios between different material zones, ill-conditioning is caused by these large stiffness ratios. Since Jacobi preconditioning operates on degrees-of-freedom, it effectively homogenizes the different spatial sub-domains. The undrained problem, modelled as a nearly incompressible problem, is much more resistant to Jacobi preconditioning, because its ill-conditioning arises from the large stiffness ratios between volumetric and distortional deformational modes, many of which involve the similar spatial domains or sub-domains. The consolidation problem has two sets of degrees-of-freedom, namely displacement and pore pressure. Some of the eigenvalues are displacement dominated whereas others are excess pore pressure dominated. Jacobi preconditioning compresses the displacement-dominated eigenvalues in a similar manner as the drained problem, but pore-pressure-dominated eigenvalues are often over-scaled. Convergence can be accelerated if this over-scaling is recognized and corrected for. Copyright © 2002 John Wiley and Sons, Ltd.