An optimal preconditioner for a class of saddle point problems with a penalty term

SIAM Journal on Scientific Computing

Volumn 19, Issue 2, 1998, Pages 540-552

  Klawonn, Axel ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

Domain decomposition; Elasticity; Mixed finite elements; Multigrid; Nearly incompressible materials; Penalty term; Preconditioned conjugate residual method; Saddle point problems

Indexed keywords

EID: 11744343493     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827595279575     Document Type: Article

References (47)

1. K. J. ARROW, L. HURWICZ, AND H. UZAWA, Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, CA, 1958.
2. S. F. ASHBY, T. A. MANTEUFFEL, AND P. E. SAYLOR, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542-1568.
3. I. BABUŠKA AND M. SURI, Locking effects in the finite element approximation of elasticity problems, Numer. Math., 62 (1992), pp. 439-463.
4. I. BABUŠKA, The finite element method with Lagrange multipliers, Numer. Math., 20 (1973), pp. 179-192.
5. R. E. BANK, B. D. WELFERT, AND H. YSERENTANT, A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1990), pp. 645-666.
6. D. BRAESS, A multigrid method for the membrane problem, Comput. Mech., 3 (1988), pp. 321-329.
7. D. BRAESS, Finite Elemente, Springer-Verlag, Berlin, 1992.
8. D. BRAESS AND C. BLÖMER, A multigrid method for a parameter dependent problem in solid mechanics, Numer. Math., 57 (1990), pp. 747-761.
9. J. H. BRAMBLE AND J. E. PASCIAK, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp., 50 (1988), pp. 1-17.
10. J. H. BRAMBLE, J. E. PASCIAK, AND A. E. VASILEV, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072-1092.
11. S. C. BRENNER, A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity, SIAM J. Numer. Anal., 30 (1993), pp. 116-135.
12. S. C. BRENNER, A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. Comp., 63 (1994), pp. 435-460.
13. S. C. BRENNER, Multigrid methods for parameter dependent problems, RAIRO Modél. Math. Anal. Numer., 30 (1996), pp. 265-297.
14. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
15. F. BREZZI, On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal. Numer., 8 (1974), pp. 129-151.
16. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
17. T. F. CHAN AND T. P. MATHEW, Domain decomposition algorithms, Acta Numerica, (1994), pp. 61-143.
18. M. DRYJA, M. SARKIS, AND O. B. WIDLUND, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math., 72 (1996), pp. 313-348.
19. M. DRYJA, B. F. SMITH, AND O. B. WIDLUND, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662-1694.
20. M. DRYJA AND O. B. WIDLUND, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-155.
21. H. ELMAN, Multigrid and Krylov subspace methods for the discrete Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), pp. 755-770.
22. H. ELMAN AND G. GOLUB, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645-1661.
23. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, New York, 1986.
24. M. GRIEBEL, Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen, Teubner Skr. Numer., Teubner-Verlag, Stuttgart, 1994.
25. M. GRIEBEL AND P. OSWALD, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., 70 (1995), pp. 163-180.
26. W. D. GROPP AND B. F. SMITH, Scalable, extensible, and portable numerical libraries, in Proc. Scalable Parallel Libraries Conference, IEEE, Los Alamitos, CA, 1994, pp. 87-93.
27. W. HACKBUSCH, Iterative Solution of Large Sparse Systems of Equations, Springer-Verlag, New York, 1994.
28. T. J. R. HUGHES, The Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1987.
29. M. JUNG, Konvergenzfaktoren von Mehrgitterverfahren für Probleme der ebenen, linearen Elastizitätstheorie, Z. Angew. Math. Mech., 67 (1987), pp. 165-173.
30. D. G. LUENBERGER, The conjugate residual method for constrained minimization problems, SIAM J. Numer. Anal., 7 (1970), pp. 390-398.
31. P. OSWALD, Multilevel Finite Element Approximation. Theory and Applications, Teubner Skr. Numer., Teubner Verlag, Stuttgart, 1994.
32. C. C. PAIGE AND M. A. SAUNDERS, Solutions of sparse linear systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
33. E. M. RØNQUIST, A domain decomposition solver for the incompressible Navier-Stokes equations, in 1994 Workshop on Spectral Element Methods, North Carolina State University.
34. T. RUSTEN AND R. WINTHER, Mixed finite element methods and domain decomposition, in Computational Methods in Water Resources IV, Volume I: Numerical Methods in Water Resources, T. F. Russel, R. E. Ewing, C. A. Brebbia, W. G. Gray, and G. F. Pinder, eds., Computational Mechanics Publications, Southhampton, UK, 1992, pp. 597-604.
35. T. RUSTEN AND R. WINTHER, A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887-904.
36. M. SARKIS, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proc. Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., Hampton, VA, 1993, NASA, number 3224, pp. 543-566.
37. M. SARKIS, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements, Ph.D. thesis, Courant Institute of the Mathematical Sciences, New York University, New York, 1994.
38. D. SILVESTER AND A. WATHEN, Fast iterative solutions of stabilised Stokes systems Part II: Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352-1367.
39. M. D. SKOGEN, Schwarz Methods and Parallelism, Ph.D. thesis, Department of Informatics, University of Bergen, Bergen, Norway, 1992.
40. G. L. G. SLEIJPEN, H. A. VAN DER VORST, AND D. R. FOKKEMA, BiCGstab(l) and other hybrid Bi-CG methods, Numer. Algorithms, 7 (1994), pp. 75-109.
41. B. F. SMITH, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, Ph.D. thesis, Courant Institute of the Mathematical Sciences, New York University, New York, 1990.
42. B. F. SMITH, A parallel implementation of an iterative sub structuring algorithm for problems in three dimensions, SIAM J. Sci. Comput., 14 (1993), pp. 406-423.
43. B. F. SMITH, Extensible PDE Solvers Package Users Manual, Tech. report ANL 94/40, Argonne National Laboratory, Argonne, IL, September 1994.
44. R. VERFÜRTH, Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO Numer. Anal., 18 (1984), pp. 175-182.
45. A. WATHEN AND D. SILVESTER, Fast iterative solution of stabilised Stokes systems Part I: Using simple diagonal preconditioners, SIAM J. Numer. Anal., 30 (1993), pp. 630-649.
46. J. XU, Theory of Multilevel Methods, Ph.D. thesis, Cornell University, Ithaca, NY, 1989.
47. X. ZHANG, Multilevel Schwarz methods, Numer. Math., 63 (1992), pp. 521-539.