GMRES convergence analysis for a convection-diffusion model problem

SIAM Journal on Scientific Computing

Volumn 26, Issue 6, 2005, Pages 1989-2009

  Liesen, J ^a   Strakos Z ^b,c  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^b INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

^c TECHNICAL UNIVERSITY OF LIBEREC   (Czech Republic)

Author keywords

Convection diffusion problem; GMRES; Ill conditioned eigenvectors; Nonnormality; Rate of convergence; Streamline upwind Petrov Galerkin discretization; Tridiagonal Toeplitz matrices

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; DIFFUSION; EIGENVALUES AND EIGENFUNCTIONS; GALERKIN METHODS; HEAT CONVECTION; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; MATRIX ALGEBRA; PROBLEM SOLVING;

CONVECTION-DIFFUSION PROBLEM; GMRES; III-CONDITIONED EIGENVECTORS; NONNORMALITY; RATE OF CONVERGENCE; STREAMLINE UPWIND PETROV-GALERKIN DISCRETIZATION; TRIDIGONAL TOEPLITZ MATRICES;

COMPUTATIONAL COMPLEXITY;

EID: 20044388272     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827503430746     Document Type: Article

References (30)

1. M. ARIOLI, V. PTÁK, AND Z. STRAKOŠ, Krylov sequences of maximal length and convergence of GMRES, BIT, 38 (1998), pp. 636-643.
2. O. AXELSSON, V. EIJKHOUT, B. POLMAN, AND P. VASSILEVSKI, Incomplete block-matrix factorization iterative methods for convection-diffusion problems, BIT, 29 (1989), pp. 867-889.
3. B. BECKERMANN AND A. KUIJLAARS, Superlinear CG convergence for special right-hand sides, Electron. Trans. Numer. Anal., 14 (2002), pp. 1-19.
4. A. N. BROOKS AND T. J. R. HUGHES, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), pp. 199-259.
5. M. EIERMANN, Semiiterative Verfahren für nichtsymmetrische lineare Gleichungssysteme, Habilitationsschrift, Universität Karlsruhe, Karlsruhe, Germany, 1989.
6. M. EIERMANN AND O. G. ERNST, GMRES and Jordan Blocks, unpublished notes, 2002.
7. H. C. ELMAN AND M. P. CHERNESKY, Ordering effects on relaxation methods applied to the discrete one-dimensional convection-diffusion equation, SIAM J. Numer. Anal., 30 (1993), pp. 1268-1290.
8. H. C. ELMAN AND A. RAMAGE, A characterization of oscillations in the discrete two-dimensional convection-diffusion equation, Math. Comp., 72 (2001), pp. 263-288.
9. H. C. ELMAN AND A. RAMAGE, An analysis of smoothing effects of upwinding strategies for the convection-diffusion equation, SIAM J. Numer. Anal., 40 (2002), pp. 254-281.
10. H. C. ELMAN, D. J. SILVESTER, AND A. J. WATHEN, Iterative methods for problems in computational fluid dynamics, in Iterative Methods in Scientific Computing (Hong Kong, 1995), Springer, Singapore, 1997, pp. 271-327.
11. M. EMBREE, GMRES Residual Behavior in a Lousy Basis, unpublished notes, 2000.
12. M. EMBREE, GMRES Residual Behavior in the Eigenvector Basis: A Study of Two Practical Examples, unpublished notes, 2000.
13. O. G. ERNST, Residual-minimizing Krylov subspace methods for stabilized discretizations of convection-diffusion equations, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1079-1101.
14. B. FISCHER, A. RAMAGE, D. SILVESTER, AND A. WATHEN, On parameter choice and iterative convergence for stabilised discretisations of advection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 179 (1999), pp. 179-195.
15. A. GREENBAUM, Some theoretical results derived from polynomial numerical hulls of Jordan blocks, Electron. Trans. Numer. Anal., submitted.
16. A. GREENBAUM AND Z. STRAKOŠ, Matrices that generate the same Krylov residual spaces, in Recent Advances in Iterative Methods, IMA Vol. Math. Appl. 60, G. H. Golub, A. Greenbaum, and M. Luskin, eds., Springer-Verlag, New York, 1994, pp. 95-118.
17. T. J. R. HUGHES AND A. BROOKS, A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), AMD 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979, pp. 19-35.
18. J. LIESEN AND Z. STRAKOŠ, Convergence Analysis of GMRES for the SUPG Discretized Convection Diffusion Model Problem, Technical report 26-2003, Technical University of Berlin, Institute of Mathematics, 2003; also available online from http://www.math.tu-berlin.de/preprints/abstracts/Report-26-2003.rdf. html.
19. J. LIESEN AND Z. STRAKOŠ, Convergence of GMRES for tridiagonal Toeplitz matrices, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 233-251.
20. J. LIESEN AND P. TICHÝ, The worst-case GMRES for normal matrices, BIT, 44 (2004), pp. 79-98.
21. K. W. MORTON, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 1996.
22. C. C. PAIGE AND Z. STRAKOŠ, Residual and backward error bounds in minimum residual Krylov subspace methods, SIAM J. Sci. Comput., 23 (2002), pp. 1898-1923.
23. A. QUARTERONI AND A. VALU, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer-Verlag, Berlin, 1994.
24. G. D. RAITHBY, Skew upstream differencing schemes for problems involving fluid flow, Comput. Methods Appl. Mech. Engrg., 9 (1976), pp. 153-164.
25. A. RAMAGE, A Note on Parameter Choice and Iterative Convergence for Stabilised Discretisations of Advection-Diffusion Problems in Three Dimensions, Technical report 32, University of Strathclyde, Department of Mathematics, 1998.
26. S. C. REDDY AND L. N. TREFETHEN, Pseudospectra of the convection-diffusion operator, SIAM J. Appl. Math., 54 (1994), pp. 1634-1649.
27. H.-G. ROOS, M. STYNES, AND L. TOBISKA, Numerical Methods for Singularly Perturbed Differential Equations, Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin, 1996.
28. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
29. G. D. SMITH, Numerical Solution of Partial Differential Equations. Finite Difference Methods, 2nd ed., Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1978.
30. L. N. TREFETHEN, Pseudospectra of linear operators, SIAM Rev., 39 (1997), pp. 383-406.