Smoothed Aggregation Multigrid for the Discontinuous Galerkin Method

The aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous Galerkin method. The smoothed aggregation multigrid, introduced by Vanek for the conforming finite element method, is applied to low-order discretizations of convection-diffusion equations. For the elliptic model problem the algorithm is shown to be quasi-optimal. Adjustments for the case of nonvanishing advection, such as directionally implicit smoothing and a suitable splitting of the operator, are discussed. Several numerical experiments are presented for two-dimensional problems, including a Newton-type linearization of the compressible Navier-Stokes equations.

[1]  Marian Brezina,et al.  Convergence of algebraic multigrid based on smoothed aggregation , 1998, Numerische Mathematik.

[2]  J. Kraus,et al.  An agglomeration-based multilevel-topology concept with application to 3D-FE meshes , 2004 .

[3]  Pieter W. Hemker,et al.  Fourier two‐level analysis for discontinuous Galerkin discretization with linear elements , 2004, Numer. Linear Algebra Appl..

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

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

[6]  Piet Hemker,et al.  On the order of prolongations and restrictions in multigrid procedures , 1990 .

[7]  M. Carbonaro,et al.  von Karman Institute for Fluid Dynamics , 2004 .

[8]  Petr Vanek,et al.  Analysis of an algebraic Petrov--Galerkin smoothed aggregation multigrid method , 2008 .

[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]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[12]  Joseph E. Pasciak,et al.  THE ANALYSIS OF SMOOTHERS FOR MULTIGRID ALGORITHMS , 1992 .

[13]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

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

[15]  David L. Darmofal,et al.  Algebraic multigrid for stabilized finite element discretizations of the Navier?Stokes equations , 2004 .

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

[17]  Vivek Sarin,et al.  Domain Decomposition , 2011, Encyclopedia of Parallel Computing.

[18]  曹志浩,et al.  ON ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS , 1993 .

[19]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[20]  Jim E. Jones,et al.  AMGE Based on Element Agglomeration , 2001, SIAM J. Sci. Comput..

[21]  Markus Wabro,et al.  AMGe - Coarsening Strategies and Application to the Oseen Equations , 2005, SIAM J. Sci. Comput..

[22]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .

[23]  C. Lasser,et al.  Overlapping preconditioners for discontinuous Galerkin approximations of second order problems , .

[24]  Hamilton-Jacobi Equations,et al.  Multigrid Methods for , 2011 .

[25]  Dimitri J. Mavriplis,et al.  Agglomeration multigrid for two-dimensional viscous flows , 1995 .

[26]  Khosro Shahbazi,et al.  An explicit expression for the penalty parameter of the interior penalty method , 2022 .

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

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

[29]  Petr Vanek,et al.  An Aggregation Multigrid Solver for convection-diffusion problems onunstructured meshes. , 1998 .

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

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

[32]  R. Hartmann,et al.  Symmetric Interior Penalty DG Methods for the CompressibleNavier-Stokes Equations I: Method Formulation , 2005 .

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

[34]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[35]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[36]  J. Kraus.  MULTILEVEL PRECONDITIONING OF ELLIPTIC PROBLEMS DISCRETIZED BY A CLASS OF DISCONTINUOUS GALERKIN METHODS , 2006 .

[37]  Ralf Hartmann,et al.  Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation , 2005 .

[38]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[39]  Marian Brezina,et al.  Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems , 2005, Computing.

[40]  Dimitri J. Mavriplis,et al.  MULTIGRID TECHNIQUES FOR UNSTRUCTURED MESHES , 1995 .

[41]  Jinchao Xu,et al.  Convergence estimates for multigrid algorithms without regularity assumptions , 1991 .

[42]  Douglas Enright,et al.  Mathematical and computational methods for compressible flow , 2005, Math. Comput..