Auxiliary Space Multigrid Method Based on Additive Schur Complement Approximation for Graph Laplacian

In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The construction is based on a two-level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non-overlapping subdomains. The two-level method utilizes a coarse-grid operator obtained from additive Schur complement approximation (ASCA). Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both, the two-level preconditioner, as well as for the ASCA are derived. The two-level method is recursively extended to define the ASMG algorithm. In particular, so-called Krylov-cycles are considered. The theoretical results are supported by a representative collection of numerical tests which further demonstrate the efficiency of the new algorithm for multiscale problems.

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