Towards automatic multigrid algorithms for SPD, nonsymmetric and indefinite problems

SIAM Journal on Scientific Computing

Volumn 17, Issue 2, 1996, Pages 439-453

  Shapira, Yair ^a   Israeli, Moshe ^a   Sidi, Avram ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Automatic multigrid method; Convection diffusion equation; Discontinuous coefficients; Elliptic PDEs; Indefinite Helmholtz equation

Indexed keywords

EID: 0043018476     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827593258838     Document Type: Article

References (34)

1. R. ALCOUFFE, A. BRANDT, J. E. DENDY, AND J. PAINTER, The multigrid method for the diffusion equation with strongly discontinuous doefficients, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 430-454.
2. P. AMODIO AND F. MAZZIA, Backward error analysis of cyclic reduction for the solution of tridiagonal systems, Math. Comp., 62 (1994), pp. 601-617.
3. J. H. BRAMBLE, Z. LEYK, AND J. E. PASCIAK, Iterative schemes for non-symmetric and indefinite elliptic boundary value problems, Math. Comp., 60 (1993), pp. 1-22.
4. A. BRANDT, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), pp. 333-390.
5. _, Guide to Multigrid Development, Multigrid Methods, W. Hackbusch and U. Trottenberg, eds., Lecture Notes in Mathematics 960, Springer-Verlag, Berlin, Heidelberg, 1982.
6. _, Algebraic multigrid theory: The symmetric case, Appl. Math. Comp., 19 (1986), pp. 23-56.
7. A. BRANDT AND S. TA'ASAN, Multigrid methods for nearly singular and slightly indefinite problems, in Multigrid Methods II, Proceedings, Cologne, 1985, Lecture Notes in Mathematics 1228, W. Hackbusch and U. Trottenberg, eds., Springer-Verlag, Berlin, pp. 100-122.
8. A. BRANDT AND I. YAVNEH, Inadequacy of first-order upwind difference schemes for some recirculating flows, J. Comp. Phys., 93 (1991), pp. 128-143.
9. _, On multigrid solution of high Reynolds incompressible entering flows, J. Comp. Phys., 101, (1992), pp. 151-164.
10. A. BRANDT AND I. YAVNEH, Accelerated multigrid convergence and high Reynolds recirculating flows, SIAM J. Sci. Statist. Comput., 14 (1993), pp. 607-626.
11. J. E. DENDY, Black box multigrid, J. Comp. Phys., 48 (1982), pp. 366-386.
12. _, Black box multigrid for nonsymmetric problems, Appl. Math. Comp., 13 (1983), pp. 261-283.
13. B. ENGQUIST AND E. LUO, Multigrid methods for differential equations with highly oscillating coefficients, Sixth Copper Mountain Conference on Multigrid Methods, N. D. Melson, S. F. McCormick, and T. A. Manteuffel, eds., NASA Langley Research Center, Hampton, VA, 1993, pp. 175-190.
14. R. W. FREUND, Conjugate gradients type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Comput., 13 (1992), pp. 425-448.
15. _, Transpose free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14 (1993), pp. 470-482.
16. R. KETTLER, Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradients methods, in Multigrid Methods, Lecture Notes in Mathematics 960, W. Hackbusch and U. Trottenberg, eds., Springer-Verlag, Berlin, Heidelberg, 1982.
17. R. KETTLER AND J. A. MEIJERINK, A Multigrid Method and a Combined Multigrid-Conjugate Gradient Method for Elliptic Problems with Strongly Discontinuous Coefficients in General Domains, Shell Publ. 604, KSELP, Rijswijk, the Netherlands, 1981.
18. J. A. MEIJERINK AND H. A. VAN DER VORST, Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems, J. Comp. Phys., 44 (1981), pp. 134-155.
19. A. QUARTERONI AND E. ZAMPIERI, Finite element preconditionering for Legendre spectral collocation approximations to elliptic equation and systems, SIAM J. Numer. Anal., 29 (1992), pp. 917-936.
20. J. W. RUGE AND K. STUBEN, Efficient solution of finite difference and finite element equations by algebraic multigrid, in Multigrid Methods for Integral and Differential Equations, D. J. Paddon and H. Holstein, eds., Oxford Univ. Press., New York, 1985, pp. 169-212.
21. _, Algebraic Multigrid, in Multigrid Methods, S. F. McCormick, ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987.
22. Y. SHAPIRA, Two-Level Analysis of Automatic Multigrid for SPD, Non-Normal and Indefinite Problems, Tech. Rep. #824, Computer Science Department, Technion-Israel Institute of Technology, July 1994; Numer. Math., submitted.
23. _, Multigrid Techniques for 3-D Definite and Indefinite Problems with Discontinuous Coefficients, Tech. Rep. #834, Computer Science Department, Technion-Israel Institute of Technology, Oct. 1994; Math. Comp., submitted.
24. _, Multigrid Techniques for Highly Indefinite Equations, 8th Copper Mountain Conference on Multigrid Methods, S. F. McCormick and T. A. Manteuffel, eds., NASA Langley Research Center, Hampton, VA, 1995, in press.
25. Y. SHAPIRA, M. ISRAELI, AND A. SIDI, An automatic multigrid method for the solution of sparse linear systems, Sixth Copper Mountain Conference on Multigrid Methods, N. D. Melson, S. F. McCormick, and T. A. Manteuffel, eds., NASA Langley Research Center, Hampton, VA, 1993, pp. 567-582.
26. R. A. SMITH AND A. WEISER, Semicoarsening multigrid on a hypercube, SIAM J. Sci. Comput., 13 (1992), pp. 1314-1329.
27. P. SONNEVELD, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36-52.
28. P. SONNEVELD, P. WESSELING, AND P. M. DE ZEEUW, Multigrid and conjugate gradient methods as convergence acceleration techniques, in Multigrid Methods for Integral and Differential Equations, D. J. Paddon and H. Holstein, eds., Oxford Univ. Press, U.K., 1985, pp. 117-168.
29. (29] H.A. VAN DER VORST, Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems, J. Comp. Phys., 49 (1982), pp. 1-19.
30. R. VARGA, Matrix Iterative Analysis, Prentice-Hall, NJ, 1962.
31. P. WESSELING, A robust and efficient multigrid solver, Multigrid Methods, W. Hackbusch and U. Trottenberg, eds., Lecture Notes in Mathematics 960, Springer-Verlag, Berlin, Heidelberg, 1982, pp. 614-630.
32. I. YAVNEH, On Red Black SOR Smoothing in Multigrid, SIAM J. Sci. Comput., submitted.
33. D. YOUNG, Iterative Solution of Large Linear Systems, Academic Press, NY, 1971.
34. P. M. DE ZEEUW, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math., 33 (1990), pp. 1-27.