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 wellbehaved 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. Copyright of XFEM, when applied to problems in linear elastic fracture mechanics, is to resolve the complexities associated with modeling of cracks by using a mesh that is independent of the crack geometry. The discontinuities along the crack interface and singularities near the crack tip are instead captured through an "enriched" space of basis functions that model the underlying physics of the fracture problem. These additional enrichment functions have local support near a crack, satisfy a partition of unity, and add to the number of degrees of freedom at the nodes near the crack. The use of an enriched space of basis functions alleviates the need for remeshing the domain in the case of propagating cracks. The crack tip enrichment functions also model singularities without the need for special elements or mesh refinement. This technique has gained a wide acceptance in the scientific community during the past decade.Concurrent with the development of computational modeling tools, numerical solvers too have evolved considerably during the past few decades. Currently, the most advanced solvers for sparse linear systems, such as those obtained from finite element discretizations, are based on building a basis for the Krylov subspace associated with the linear system and finding an approximate solution iteratively in this span. There are various preconditioning schemes available to accelerate the Krylov subspace solvers, and well-known among them is the multigrid method [7][8][9][10]. This method is based on generating a hierarchy of discretizations for the problem in order to resolve and smooth the error at different modes, thereby improving the convergence properties of the solver. Whereas geometric multigrid methods have been in use for a long time [11,12], the state-of-the-art in this field are the algebraic ...
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