Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations

We present iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems. We analyze the convergence properties of these algorithms for both symmetric and non-symmetric IP schemes. The iterative methods are based on a “natural” decomposition of the first order DG finite element space as a direct sum of the Crouzeix-Raviart non-conforming finite element space and a subspace that contains functions discontinuous at interior faces. We also present numerical examples confirming the theoretical results.

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

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

[3]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[4]  Susanne C. Brenner,et al.  A weakly over-penalized symmetric interior penalty method for the biharmonic problem. , 2010 .

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

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

[7]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[8]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[9]  M. Wheeler.  An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .

[10]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[11]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[12]  Mary F. Wheeler,et al.  Compatible algorithms for coupled flow and transport , 2004 .

[13]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[14]  Svetozar Margenov,et al.  On the multilevel preconditioning of Crouzeix-Raviart elliptic problems , 2008, Numer. Linear Algebra Appl..

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

[16]  Blanca Ayuso de Dios,et al.  Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems , 2009, SIAM J. Numer. Anal..

[17]  Mary F. Wheeler,et al.  Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media , 2005, SIAM J. Numer. Anal..

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

[19]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

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

[21]  Susanne C. Brenner,et al.  A Weakly Over-Penalized Non-Symmetric Interior Penalty Method , 2007 .

[22]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[23]  P. C. Kearney,et al.  Prepared for publication by , 1985 .

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

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

[26]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[27]  Roland Glowinski,et al.  Computing Methods in Applied Sciences , 1977 .

[28]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[29]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

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

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

[32]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

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

[34]  G. Strang.  VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD , 1972 .

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

[36]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[37]  Susanne C. Brenner,et al.  Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems , 2003, Math. Comput..

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

[39]  Ludmil T. Zikatanov,et al.  On two‐grid convergence estimates , 2005, Numer. Linear Algebra Appl..

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

[41]  Susanne C. Brenner,et al.  Convergence of nonconforming multigrid methods without full elliptic regularity , 1999, Math. Comput..

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