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Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures

Authors

  • Maksymilian Dryja,

    1. Department of Mathematics, Warsaw University, Warsaw 02-097, Poland
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  • Juan Galvis,

    1. Department of Mathematics, Texas A&M University, College Station, Texas 3368
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  • Marcus Sarkis

    Corresponding author
    1. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, Rio de Janeiro, Brazil
    2. Department of Mathematical Sciences at Worcester Polytechnic Institute, Worcester, Massachusetts 01609
    • Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, Rio de Janeiro, Brazil
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Abstract

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation equation image is introduced. To handle the nonmatching meshes across Ωi, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann-Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across Ωi, a condition number estimate equation image is established with C independent of hi, Hi, hi/hj, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

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