Preconditioning in H$(\operatorname {div})$ and applications
HTML articles powered by AMS MathViewer
- by Douglas N. Arnold, Richard S. Falk and R. Winther PDF
- Math. Comp. 66 (1997), 957-984 Request permission
Abstract:
We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I$- \operatorname {\mathbf {grad}}\operatorname {div}$. The natural setting for such problems is in the Hilbert space H$(\operatorname {div})$ and the variational formulation is based on the inner product in H$(\operatorname {div})$. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.References
- D. N. Arnold, R. S. Falk, and R. Winther, Preconditioning discrete approximations of the Reissner-Mindlin plate model, Preprint (1995).
- D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the $V$-cycle, SIAM J. Numer. Anal. 20 (1983), no. 5, 967–975. MR 714691, DOI 10.1137/0720066
- James H. Bramble, Multigrid methods, Pitman Research Notes in Mathematics Series, vol. 294, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1247694
- James H. Bramble and Joseph E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), no. 181, 1–17. MR 917816, DOI 10.1090/S0025-5718-1988-0917816-8
- James H. Bramble and Joseph E. Pasciak, New estimates for multilevel algorithms including the $V$-cycle, Math. Comp. 60 (1993), no. 202, 447–471. MR 1176705, DOI 10.1090/S0025-5718-1993-1176705-9
- J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of inexact Uzawa algorithm for saddle point problems, to appear in SIAM J. Numer. Anal. 34 (1997).
- James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), no. 195, 1–21. MR 1090464, DOI 10.1090/S0025-5718-1991-1090464-8
- James H. Bramble, Joseph E. Pasciak, and Jinchao Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), no. 193, 1–34. MR 1052086, DOI 10.1090/S0025-5718-1991-1052086-4
- Susanne C. Brenner, A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method, SIAM J. Numer. Anal. 29 (1992), no. 3, 647–678. MR 1163350, DOI 10.1137/0729042
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
- Zhi Qiang Cai, Charles I. Goldstein, and Joseph E. Pasciak, Multilevel iteration for mixed finite element systems with penalty, SIAM J. Sci. Comput. 14 (1993), no. 5, 1072–1088. MR 1232176, DOI 10.1137/0914065
- Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations. I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785–1799. MR 1302685, DOI 10.1137/0731091
- Z. Chen, Equivalence between and multigrid algorithms for mixed and nonconforming methods for second order elliptic problems, East-West J. Numer. Math. 4 (1996).
- Zhangxin Chen, Richard E. Ewing, and Raytcho Lazarov, Domain decomposition algorithms for mixed methods for second-order elliptic problems, Math. Comp. 65 (1996), no. 214, 467–490. MR 1333307, DOI 10.1090/S0025-5718-96-00703-X
- L. C. Cowsar, Dual-variable Schwarz methods for mixed finite elements, Report TR93–09, Rice University (1993).
- Lawrence C. Cowsar, Jan Mandel, and Mary F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64 (1995), no. 211, 989–1015. MR 1297465, DOI 10.1090/S0025-5718-1995-1297465-9
- Maksymilian Dryja and Olof B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 3–21. MR 1064335
- M. Dryja and O. B. Widlund, Some domain decomposition algorithms for elliptic problems, Iterative methods for large linear systems, L. Hayes and D. Kincaid, eds., Academic Press, San Diego, 1990, pp. 273-291.
- Howard C. Elman and Gene H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), no. 6, 1645–1661. MR 1302679, DOI 10.1137/0731085
- R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753, DOI 10.1051/m2an/1980140302491
- Roland Glowinski and Mary Fanett Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 144–172. MR 972516
- P. Lin, A sequential regularization method for time-dependent incompressible Navier–Stokes equations, to appear in SIAM J. Numer. Anal. 34 (1997).
- Tarek P. Mathew, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems. I. Algorithms and numerical results, Numer. Math. 65 (1993), no. 4, 445–468. MR 1231895, DOI 10.1007/BF01385762
- A. I. Pehlivanov, G. F. Carey, and R. D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1368–1377. MR 1293520, DOI 10.1137/0731071
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
- Torgeir Rusten and Ragnar Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 3, 887–904. Iterative methods in numerical linear algebra (Copper Mountain, CO, 1990). MR 1168084, DOI 10.1137/0613054
- Torgeir Rusten and Ragnar Winther, Substructure preconditioners for elliptic saddle point problems, Math. Comp. 60 (1993), no. 201, 23–48. MR 1149293, DOI 10.1090/S0025-5718-1993-1149293-0
- Torgeir Rusten, Panayot S. Vassilevski, and Ragnar Winther, Interior penalty preconditioners for mixed finite element approximations of elliptic problems, Math. Comp. 65 (1996), no. 214, 447–466. MR 1333325, DOI 10.1090/S0025-5718-96-00720-X
- T. Rusten, P. S. Vassilevski, and R. Winther, Domain embedding preconditioners for mixed systems, in preparation.
- P. Vassilevski and R. Lazarov, Preconditioning saddle-point problems arising from mixed finite element discretizations of elliptic equations, to appear in Numer. Lin. Algebra Appl.
- Panayot S. Vassilevski and Junping Wang, An application of the abstract multilevel theory to nonconforming finite element methods, SIAM J. Numer. Anal. 32 (1995), no. 1, 235–248. MR 1313711, DOI 10.1137/0732008
- Panayot S. Vassilevski and Jun Ping Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), no. 4, 503–520. MR 1189534, DOI 10.1007/BF01385872
- Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Additional Information
- Douglas N. Arnold
- Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 27240
- Email: dna@math.psu.edu
- Richard S. Falk
- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- Email: falk@math.rutgers.edu
- R. Winther
- Affiliation: Department of Informatics, University of Oslo, Oslo, Norway
- MR Author ID: 183665
- Email: ragnar@ifi.uio.no
- Received by editor(s): March 19, 1996
- Received by editor(s) in revised form: April 19, 1996
- Additional Notes: The first author was supported by NSF grants DMS-9205300 and DMS-9500672 and by the Institute for Mathematics and its Applications. The second author was supported by NSF grant DMS-9403552. The third author was supported by The Norwegian Research Council under grants 100331/431 and STP.29643.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 957-984
- MSC (1991): Primary 65N55, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-97-00826-0
- MathSciNet review: 1401938
Dedicated: Dedicated to Professor Ivo Babuška on the occasion of his seventieth birthday.