Two-Level Overlapping Schwarz Algorithms for a Staggered Discontinuous Galerkin Method

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [SIAM J. Numer. Anal., 47 (2009), pp. 3820--3848] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.

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

[2]  Xiaobing Feng,et al.  Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation , 2005, J. Sci. Comput..

[3]  B. Bojarski,et al.  Remarks on Sobolev imbedding inequalities , 1988 .

[4]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

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

[6]  Olof B. Widlund,et al.  Two-Level Schwarz Algorithms with Overlapping Subregions for Mortar Finite Elements , 2006, SIAM J. Numer. Anal..

[7]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

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

[9]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

[10]  Pekka Koskela,et al.  Sobolev-Poincaré implies John , 1995 .

[11]  Eric T. Chung,et al.  A staggered discontinuous Galerkin method for the curl–curl operator , 2012 .

[12]  Paola F. Antonietti,et al.  Two-Level Schwarz Preconditioners for Super Penalty Discontinuous Galerkin Methods , 2007 .

[13]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions , 2009, SIAM J. Numer. Anal..

[14]  Olof B. Widlund,et al.  Accomodating Irregular Subdomains in Domain Decomposition Theory , 2009 .

[15]  Susanne C. Brenner,et al.  Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method , 2011, J. Sci. Comput..

[16]  F. Brezzi,et al.  Discontinuous Galerkin approximations for elliptic problems , 2000 .

[17]  Clark R. Dohrmann,et al.  An Iterative Substructuring Algorithm for Two-Dimensional Problems in H(curl) , 2012, SIAM J. Numer. Anal..

[18]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

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

[20]  Olof B. Widlund,et al.  An Alternative Coarse Space for Irregular Subdomains and an Overlapping Schwarz Algorithm for Scalar Elliptic Problems in the Plane , 2011, SIAM J. Numer. Anal..

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

[22]  Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptical Problems Part I. Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptic Problems , 1999 .

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

[24]  Jun Zou,et al.  Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids , 1996 .

[25]  Benjamin Stamm,et al.  Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems , 2008, SIAM J. Numer. Anal..

[26]  Paul Houston,et al.  A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods , 2011, J. Sci. Comput..

[27]  Eric T. Chung,et al.  A staggered discontinuous Galerkin method for the convection–diffusion equation , 2012, J. Num. Math..

[28]  S. C. Brenner,et al.  POINCAR´ E-FRIEDRICHS INEQUALITIES FOR PIECEWISE H 1 FUNCTIONS ∗ , 2003 .

[29]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[30]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for Wave Propagation , 2006, SIAM J. Numer. Anal..

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

[32]  Olof B. Widlund,et al.  Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions , 2008, SIAM J. Numer. Anal..

[33]  Olof B. Widlund,et al.  An Analysis of a FETI-DP Algorithm on Irregular Subdomains in the Plane , 2008, SIAM J. Numer. Anal..