On Hybrid Multigrid-Schwarz Algorithms

Abstract J. Lottes and P. Fischer (J. Sci. Comput. 24:45–78, [2005]) studied many smoothers or preconditioners for hybrid Multigrid-Schwarz algorithms for the spectral element method. The behavior of several of these smoothers or preconditioners are analyzed in the present paper. Here it is shown that the Schwarz smoother that best performs in the above reference, is equivalent to a special case of the weighted restricted additive Schwarz, for which convergence analysis is presented. For other preconditioners which do not perform as well, examples and explanations are presented illustrating why this behavior may occur.

[1]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[2]  S. Nepomnyaschikh,et al.  Domain Decomposition Methods , 2007 .

[3]  Daniel B. Szyld,et al.  Weighted max norms, splittings, and overlapping additive Schwarz iterations , 1999, Numerische Mathematik.

[4]  William Gropp,et al.  PETSc 2.0 users manual , 2000 .

[5]  Paul F. Fischer,et al.  Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method , 2005, J. Sci. Comput..

[6]  Charbel Farhat,et al.  A Minimum Overlap Restricted Additive Schwarz Preconditioner and Applications in 3D Flow Simulations , 1998 .

[7]  Anthony T. Patera,et al.  Spectral element multigrid. I. Formulation and numerical results , 1987 .

[8]  R. Glowinski,et al.  Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , 1990 .

[9]  Olof B. Widlund,et al.  To overlap or not to overlap: a note on a domain decomposition method for elliptic problems , 1989 .

[10]  Daniel B. Szyld,et al.  An Algebraic Convergence Theory for Restricted Additive Schwarz Methods Using Weighted Max Norms , 2001, SIAM J. Numer. Anal..

[11]  Michele Benzi,et al.  Algebraic theory of multiplicative Schwarz methods , 2001, Numerische Mathematik.

[12]  A. Patera.  A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .

[13]  Xiao-Chuan Cai,et al.  A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[14]  Daniel B. Szyld,et al.  Convergence Theory of Restricted Multiplicative Schwarz Methods , 2002, SIAM J. Numer. Anal..

[15]  Daniel B. Szyld,et al.  Schwarz Iterations for Symmetric Positive Semidefinite Problems , 2006, SIAM J. Matrix Anal. Appl..

[16]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[17]  Jacques Periaux,et al.  On Domain Decomposition Methods , 1988 .

[18]  Paul Fischer,et al.  Hybrid Schwarz-Multigrid Methods for the Spectral Element Method: Extensions to Navier-Stokes , 2005 .