A generalized predictive analysis tool for multigrid methods

Summary Multigrid and related multilevel methods are the approaches of choice for solving linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the well-known local mode (often local Fourier) analysis approach to a wider class of problems. The semi-algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two-dimensional convection-diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short-term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  Stefan Vandewalle,et al.  Space-time Concurrent Multigrid Waveform Relaxation , 1994 .

[2]  Thomas A. Manteuffel,et al.  Towards Adaptive Smoothed Aggregation (AlphaSA) for Nonsymmetric Problems , 2010, SIAM J. Sci. Comput..

[3]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[4]  Ray S. Tuminaro,et al.  A New Petrov--Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems , 2008, SIAM J. Sci. Comput..

[5]  Cornelis W. Oosterlee,et al.  Multigrid Line Smoothers for Higher Order Upwind Discretizations of Convection-Dominated Problems , 1998 .

[6]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[7]  Stefan Vandewalle,et al.  Fourier mode analysis of multigrid methods for partial differential equations with random coefficients , 2007, J. Comput. Phys..

[8]  S. McCormick,et al.  Towards Adaptive Smoothed Aggregation (αsa) for Nonsymmetric Problems * , 2022 .

[9]  J. Lions,et al.  Résolution d'EDP par un schéma en temps « pararéel » , 2001 .

[10]  James L. Thomas,et al.  Half-Space Analysis of the Defect-Correction Method for Fromm Discretization of Convection , 2000, SIAM J. Sci. Comput..

[11]  Stefan Vandewalle,et al.  Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case , 1996, SIAM J. Sci. Comput..

[12]  Stefan Vandewalle,et al.  Efficient Parallel Algorithms for Solving Initial-Boundary Value and Time-Periodic Parabolic Partial Differential Equations , 1992, SIAM J. Sci. Comput..

[13]  Stefan Vandewalle,et al.  Multigrid Waveform Relaxation for Anisotropic Partial Differential Equations , 2002, Numerical Algorithms.

[14]  Stefan Vandewalle,et al.  Local Fourier Analysis of Multigrid for the Curl-Curl Equation , 2008, SIAM J. Sci. Comput..

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

[16]  K. Stüben,et al.  Multigrid methods: Fundamental algorithms, model problem analysis and applications , 1982 .

[17]  Cornelis Vuik,et al.  A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow , 2004, SIAM J. Numer. Anal..

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

[19]  L. Trefethen.  Spectra and pseudospectra , 2005 .

[20]  Cornelis W. Oosterlee,et al.  FOURIER ANALYSIS OF GMRES ( m ) PRECONDITIONED BY MULTIGRID , 2000 .

[21]  P. Salinas,et al.  Local Fourier analysis for cell-centered multigrid methods on triangular grids , 2014, J. Comput. Appl. Math..

[22]  Cornelis W. Oosterlee,et al.  An Efficient Multigrid Solver based on Distributive Smoothing for Poroelasticity Equations , 2004, Computing.

[23]  Jose L. Gracia,et al.  Fourier Analysis for Multigrid Methods on Triangular Grids , 2009, SIAM J. Sci. Comput..

[24]  Alexander Ostermann,et al.  Multi-grid dynamic iteration for parabolic equations , 1987 .

[25]  U. Trottenberg,et al.  A note on MGR methods , 1983 .

[26]  I. Yavneh,et al.  On Multigrid Solution of High-Reynolds Incompressible Entering Flows* , 1992 .

[27]  Graham Horton,et al.  Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods , 2005, Computing.

[28]  A. Brandt,et al.  The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients , 1981 .

[29]  Ludmil T. Zikatanov,et al.  Circulant block-factorization preconditioning of anisotropic elliptic problems , 1997, Computing.

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

[31]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[32]  Cornelis W. Oosterlee,et al.  Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs , 2011, Numer. Linear Algebra Appl..

[33]  S. Sivaloganathan,et al.  The use of local mode analysis in the design and comparison of multigrid methods , 1991 .

[34]  Guohua Zhou,et al.  Fourier Analysis of Multigrid Methods on Hexagonal Grids , 2009, SIAM J. Sci. Comput..

[35]  Graham Horton,et al.  The time‐parallel multigrid method , 1992 .

[36]  J. Dendy.  Black box multigrid , 1982 .

[37]  Cornelis W. Oosterlee,et al.  On Three-Grid Fourier Analysis for Multigrid , 2001, SIAM J. Sci. Comput..

[38]  BORIS DISKIN,et al.  On Quantitative Analysis Methods for Multigrid Solutions , 2005, SIAM J. Sci. Comput..

[39]  Scott P. MacLachlan,et al.  Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes , 2013, SIAM J. Sci. Comput..

[40]  J. Meijerink,et al.  An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients , 1999 .

[41]  A. Brandt.  Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L 2 -norm , 1994 .

[42]  Graham Horton,et al.  An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations , 1995, SIAM J. Sci. Comput..

[43]  Carmen Rodrigo,et al.  Multigrid Methods on Semi-Structured Grids , 2012 .

[44]  Jürg Nievergelt,et al.  Parallel methods for integrating ordinary differential equations , 1964, CACM.

[45]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .

[46]  Ralf Knirsch,et al.  A time-parallel multigrid-extrapolation method for parabolic partial differential equations , 1992, Parallel Comput..

[47]  Ulrich Rüde,et al.  Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis , 2013, Comput. Math. Appl..

[48]  Stefan Vandewalle,et al.  Multigrid waveform relaxation on spatial finite element meshes , 1996 .

[49]  Graham Horton.  Time-Parallel Multigrid Solution of the Navier-Stokes Equations , 1991 .

[50]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[51]  A. Niestegge,et al.  Analysis of a multigrid strokes solver , 1990 .

[52]  Ivan G. Graham,et al.  Domain decomposition for multiscale PDEs , 2007, Numerische Mathematik.

[53]  Robert D. Falgout,et al.  Parallel time integration with multigrid , 2013, SIAM J. Sci. Comput..

[54]  I. Graham,et al.  Robust domain decomposition algorithms for multiscale PDEs , 2007 .

[55]  Robert D. Falgout,et al.  Parallel time integration with multigrid , 2014 .

[56]  Achi Brandt,et al.  Multigrid solvers for the non-aligned sonic flow: the constant coefficient case , 1999 .

[57]  Tobias Weinzierl,et al.  A Geometric Space-Time Multigrid Algorithm for the Heat Equation , 2012 .