Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.

[1]  Zi-Cai Li,et al.  Schwarz Alternating Method , 1998 .

[2]  M. Holst,et al.  Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients , 2010, Math. Comput..

[3]  S. C. Brenner,et al.  Convergence of Multigrid Algorithms for Interior Penalty Methods , 2005 .

[4]  Susanne C. Brenner,et al.  A Nonoverlapping DD Preconditioner for a Weakly Over-Penalized Symmetric Interior Penalty Method , 2013, Domain Decomposition Methods in Science and Engineering XX.

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

[6]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

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

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

[9]  Olof B. Widlund,et al.  Two-Level Overlapping Schwarz Algorithms for a Staggered Discontinuous Galerkin Method , 2013, SIAM J. Numer. Anal..

[10]  Claudio Canuto,et al.  BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree , 2014 .

[11]  Juan Galvis,et al.  Neumann‐Neumann methods for a DG discretization on geometrically nonconforming substructures , 2012 .

[12]  O. Widlund.  Some Schwarz Methods for Symmetric and Nonsymmetric Elliptic Problems , 1991 .

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

[14]  L. D. Marini,et al.  Stabilization mechanisms in discontinuous Galerkin finite element methods , 2006 .

[15]  Wolfgang Dahmen,et al.  A Multilevel Preconditioner for the Interior Penalty Discontinuous Galerkin Method , 2008, SIAM J. Numer. Anal..

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

[17]  L. Zikatanov,et al.  A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations , 2011, 1110.5743.

[18]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for nonconforming finite element methods , 1996, Math. Comput..

[19]  Jinchao Xu,et al.  The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids , 1996, Computing.

[20]  Ohannes A. Karakashian,et al.  A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

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

[22]  Blanca Ayuso de Dios,et al.  A Block Solver for the Exponentially Fitted IIPG-0 Method , 2013, Domain Decomposition Methods in Science and Engineering XX.

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

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

[25]  Juan Galvis,et al.  A FETI-DP Preconditioner for a Composite Finite Element and Discontinuous Galerkin Method , 2013, SIAM J. Numer. Anal..

[26]  Martin J. Gander,et al.  Block Jacobi for Discontinuous Galerkin Discretizations: No Ordinary Schwarz Methods , 2014 .

[27]  S. Nepomnyaschikh.  Mesh theorems on traces, normalizations of function traces and their inversion , 1991 .

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

[29]  Wolfgang Dahmen,et al.  Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods Part I: Geometrically Conforming Meshes , 2013 .

[30]  P. F. Antonietti,et al.  Substructuring preconditioners for an h p Nitsche-type method , 2013, 1301.3175.

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

[32]  J. Galvis,et al.  NEUMANN-NEUMANN METHODS FOR A DG DISCRETIZATION OF ELLIPTIC PROBLEMS WITH DISCONTINUOUS COEFFICIENTS ON GEOMETRICALLY NONCONFORMING SUBSTRUCTURES , 2009 .

[33]  Jinchao Xu,et al.  The method of alternating projections and the method of subspace corrections in Hilbert space , 2002 .

[34]  Peter Oswald.  Preconditioners for nonconforming discretizations , 1996, Math. Comput..

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

[36]  Christoph Lehrenfeld,et al.  Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes , 2013 .