Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods

Traditional algebraic multigrid (AMG) preconditioners are not well suited for crack problems modeled by extended finite element methods (XFEM). This is mainly because of the unique XFEM formulations, which embed discontinuous fields in the linear system by addition of special degrees of freedom. These degrees of freedom are not properly handled by the AMG coarsening process and lead to slow convergence. In this paper, we proposed a simple domain decomposition approach that retains the AMG advantages on well-behaved domains by avoiding the coarsening of enriched degrees of freedom. The idea was to employ a multiplicative Schwarz preconditioner where the physical domain was partitioned into "healthy" (or unfractured) and "cracked" subdomains. First, the "healthy" subdomain containing only standard degrees of freedom, was solved approximately by one AMG V-cycle, followed by concurrent direct solves of "cracked" subdomains. This strategy alleviated the need to redesign special AMG coarsening strategies that can handle XFEM discretizations. Numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force AMG preconditioner applied to the linear system.