Multigrid Solver for the Inner Problem in Domain Decomposition Methods forp-FEM

Author:
  • SBSven Beuchler
    Sven Beuchler
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Abstract
From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p-version finite element discretizations of elliptic boundary value problems. The ingredients of such a preconditioner are a preconditioner for the Schur complement, a preconditioner related to the Dirichlet problems in the subdomains, and an extension operator from the boundaries of the subdomains into their interior. In the case of Poisson's equation, we propose a preconditioner for the problems in the subdomains which can be interpreted as the stiffness matrix resulting from an h-version finite element discretization of a degenerate operator. For solving the corresponding systems of finite element equations a multigrid algorithm with a special line smoother is used. We prove that the convergence rate of the multigrid method is independent of the discretization parameter. The proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples.

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