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

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2-D 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 Thi(Ωi) is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. In this paper 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 C(1 + maxi log Hi hi ) 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.

[1]  R. Yates,et al.  Unified multipliers‐free theory of dual‐primal domain decomposition methods , 2009 .

[2]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[3]  Barry F. Smith,et al.  Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions , 1994 .

[4]  Xiaobing Feng,et al.  Analysis of Two-Level Overlapping Additive Schwarz Preconditioners for a Discontinuous Galerkin Method , .

[5]  Marcus Sarkis,et al.  Two-level Schwartz methods for nonconforming finite elements and discontinuous coefficients , 1993 .

[6]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for C0 interior penalty methods , 2005, Numerische Mathematik.

[7]  Andrea Toselli,et al.  An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems , 2000, Math. Comput..

[8]  Marian Brezina,et al.  Balancing domain decomposition for problems with large jumps in coefficients , 1996, Math. Comput..

[9]  Ludmil T. Zikatanov,et al.  Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations , 2006, Numer. Linear Algebra Appl..

[10]  Satyendra K. Tomar,et al.  Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods , 2008, SIAM J. Sci. Comput..

[11]  M. Sarkis.  Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements , 1997 .

[12]  Jinchao Xu,et al.  UNIFORM CONVERGENT MULTIGRID METHODS FOR ELLIPTIC PROBLEMS WITH STRONGLY DISCONTINUOUS COEFFICIENTS , 2008 .

[13]  Erik Burman,et al.  A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems , 2006, SIAM J. Numer. Anal..

[14]  O. Widlund,et al.  Schwarz Methods of Neumann-Neumann Type for Three-Dimensional Elliptic Finite Element Problems , 1993 .

[15]  Ismael Herrera-Revilla.  New formulation of iterative substructuring methods without Lagrange multipliers: Neumann–Neumann and FETI , 2008 .

[16]  Paola F. Antonietti,et al.  Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case , 2007 .

[17]  Ian G. Graham,et al.  Unstructured Additive Schwarz-Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients , 1999, SIAM J. Sci. Comput..

[18]  Olof B. Widlund,et al.  A BDDC Method for Mortar Discretizations Using a Transformation of Basis , 2008, SIAM J. Numer. Anal..

[19]  Satyendra K. Tomar,et al.  A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems , 2008, Numer. Linear Algebra Appl..

[20]  Stanimire Tomov,et al.  Interior Penalty Discontinuous Approximations of Elliptic Problems , 2001 .

[21]  Juan Galvis,et al.  BDDC methods for discontinuous Galerkin discretization of elliptic problems , 2007, J. Complex..

[22]  Xiaobing Feng,et al.  Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems , 2001, SIAM J. Numer. Anal..

[23]  M. Dryja.  On Discontinuous Galerkin Methods for Elliptic Problems with Discontinuous Coefficients , 2003 .

[24]  Guido Kanschat,et al.  A multilevel discontinuous Galerkin method , 2003, Numerische Mathematik.

[25]  Maksymilian Dryja.  On discontinuos Galerkin methods for elliptic problems with discontinuous coefficints , 2003 .

[26]  Blanca Ayuso de Dios,et al.  Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations , 2009, J. Sci. Comput..

[27]  O. Widlund,et al.  Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions , 1994 .

[28]  Guido Kanschat,et al.  Preconditioning Methods for Local Discontinuous Galerkin Discretizations , 2003, SIAM J. Sci. Comput..

[29]  Juan Galvis,et al.  Balancing Domain Decomposition Methods for Discontinuous Galerkin Discretization , 2008 .

[30]  Paola F. Antonietti,et al.  Multiplicative Schwarz Methods for Discontinuous Galerkin Approximations ofElliptic Problems , 2007 .

[31]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[32]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[33]  Svetozar Margenov,et al.  CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems , 2007, J. Complex..

[34]  J. Mandel.  Balancing domain decomposition , 1993 .