Effective Two-Level Domain Decomposition Preconditioners for Elastic Crack Problems Modeled by Extended Finite Element Method

Abstract

In this paper, we propose some effective one- and two-level domain decomposition preconditioners for elastic crack problems modeled by extended finite element method. To construct the preconditioners, the physical domain is decomposed into the "crack tip" subdomain, which contains all the degrees of freedom (dofs) of the branch enrichment functions, and the "regular" subdomains, which contain the standard dofs and the dofs of the Heaviside enrichment function. In the one-level additive Schwarz and restricted additive Schwarz preconditioners, the "crack tip" subproblem is solved directly and the "regular" subproblems are solved by some inexact solvers, such as ILU. In the two-level domain decomposition preconditioners, traditional interpolations between the coarse and the fine meshes destroy the good convergence property. Therefore, we propose an unconventional approach in which the coarse mesh is exactly the same as the fine mesh along the crack line, and adopt the technique of a non-matching grid interpolation between the fine and the coarse meshes. Numerical experiments demonstrate the effectiveness of the two-level domain decomposition preconditioners applied to elastic crack problems.