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Interior penalty preconditioners
for mixed finite element approximations
of elliptic problems


Authors: Torgeir Rusten, Panayot S. Vassilevski and Ragnar Winther
Journal: Math. Comp. 65 (1996), 447-466
MSC (1991): Primary 65F10, 65N20, 65N30
DOI: https://doi.org/10.1090/S0025-5718-96-00720-X
MathSciNet review: 1333325
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Abstract: It is established that an interior penalty method applied to second-order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example, a family of additive Schwarz preconditioners for these systems is constructed. Numerical examples which confirm the theoretical results are also presented.


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  • 1. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), pp. 742--760. MR 83f:65173
  • 2. O. Axelsson, Numerical algorithms for indefinite problems, Elliptic Problem Solvers II (G. Birkhoff and A. Schoenstadt, eds.), Academic Press, Orlando, 1984, pp. 219--232. CMP 17:03
  • 3. O. Axelsson and P. S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable--step preconditioning, SIAM J. Matrix Anal. Appl. 12 (1991), pp. 625--644. MR 92m:65046
  • 4. ------, Construction of variable--step preconditioners for inner--outer iterative methods, Iterative Methods in Linear Algebra (R. Beauwens and P. de Groen, eds.), North-Holland, Amsterdam, 1992, pp. 1--14. CMP 92:11
  • 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. MR 91b:65035
  • 6. P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23 (1986), pp. 1097--1120. MR 88h:65188
  • 7. 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. MR 89m:65097a
  • 8. ------, Iterative techniques for time dependent Stokes problem. Preprint 1994.
  • 9. J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), pp. 103--134. MR 87m:65174
  • 10. ------, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), pp. 361--369. MR 88a:65123
  • 11. J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems. Preprint 1994.
  • 12. J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), pp. 1--34. MR 91h:65159
  • 13. F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O. Mathematical Modelling and Numerical Analysis 21 (1987), no. 4, pp. 581--604. MR 88j:65249
  • 14. F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), pp. 217--235. MR 87g:65133
  • 15. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 92d:65187
  • 16. L. C. Cowsar, Dual variable Schwarz methods for mixed finite elements, Report TR93-09, Rice University, Houston, 1993.
  • 17. L. C. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64 (1995), pp. 989--1015. MR 95j:65161
  • 18. J. Douglas, Jr. and J. Wang, A new family of mixed finite element spaces over rectangles, Comp. Appl. Math. 12 (1993), pp. 183--197. CMP 94:16
  • 19. M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report 339, Courant Institute of Mathematical Sciences, 1987.
  • 20. M. Dryja, B. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, Technical Report 638, Courant Institute of Mathematical Sciences, 1993.
  • 21. H. C. Elman and G. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), pp. 1645--1661. MR 95f:65065
  • 22. R. E. Ewing and M. F. Wheeler, Computational aspects of mixed finite element methods, Numerical Methods for Scientific Computing (R. S. Stepleman, ed.), North-Holland Publishing Co., Amsterdam, 1983, pp. 163--172.
  • 23. R. S. Falk and J. E. Osborn, Error estimates for mixed methods, R.A.I.R.O. Numerical Analysis 14 (1980), no. 3, pp. 249--277. MR 82j:65076
  • 24. V. Girault and P. A. Raviart, Finite element methods for the Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 88b:65129
  • 25. R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, Proceedings, First International Conference on Domain Decomposition Methods (Philadelphia) (R. Glowinski et al., eds.), SIAM, 1988, pp. 144--172. MR 90a:65237
  • 26. P. L. Lions, On the Schwarz alternating method, Proceedings, First International Conference on Domain Decomposition Methods (Philadelphia) (R. Glowinski et al., eds.), SIAM, 1988, pp. 1--42. MR 90a:65248
  • 27. A.M. Matsokin and S. Nepomnyaschikh, On the Schwarz alternating method, Preprint, Computing Center, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1984.
  • 28. C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), pp. 617--629. MR 52:4595
  • 29. W. Queck, The convergence factor of preconditioned algorithms of the Arrow-Hurwich type, SIAM J. Numer. Anal. 26 (1989), pp. 1016--1030. MR 90m:65071
  • 30. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, nr. 606 (I. Galligani and E. Magenes, eds.), Springer Verlag, Berlin, 1977, pp. 295--315. MR 58:3547
  • 31. T. Rusten and R. Winther, A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 887--904. MR 93a:65043
  • 32. ------, Substructure preconditioners for elliptic saddle point problems, Math. Comp. 60 (1993), pp. 23--48. MR 93d:65108
  • 33. D. S. Silvester and A. J. Wathen, Fast iterative solution of stabilized Stokes systems, part II: using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), pp. 1352--1367. MR 95g:65132
  • 34. P. S. Vassilevski and R. D. Lazarov, Preconditioning saddle-point problems arising from mixed finite element discretization of elliptic problems, Report CAM 92--46, Department of Mathematics, UCLA, 1992.
  • 35. P. S. Vassilevski and J. Wang, An application of the abstract multilevel theory to nonconforming finite element methods, SIAM J. Numer. Anal. 32 (1995), 235--248. CMP 95:07
  • 36. ------, Multilevel methods for cell--centered finite difference approximations of elliptic problems, Preprint, 1992.
  • 37. R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. Numerical Analysis (1984), pp. 441--455. MR 86f:65200
  • 38. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34 (1992), pp. 581--613. MR 93k:65029

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Additional Information

Torgeir Rusten
Affiliation: SINTEF, P. O. Box 124 Blindern, N-0314 Oslo, Norway
Email: Torgeir.Rusten@si.sintef.no

Panayot S. Vassilevski
Affiliation: Center of Informatics and Computer Technology, Bulgarian Academy of Sciences, “Acad. G. Bontchev” street, Block 25 A, 1113 Sofia, Bulgaria
Email: panayot@iscbg.acad.bg

Ragnar Winther
Affiliation: Department of Informatics, University of Oslo, P. O. Box 1080 Blindern, N-0316 Oslo, Norway
Email: Ragnar.Winther@ifi.uio.no

DOI: https://doi.org/10.1090/S0025-5718-96-00720-X
Keywords: Second-order elliptic\ problems, mixed finite elements, interior penalty preconditioners, domain decomposition
Received by editor(s): August 3, 1994
Received by editor(s) in revised form: November 29, 1994
Additional Notes: The work of all authors was partially supported by the Research Council of Norway (NFR), program no. 100998/420 and STP.29643. The work of the second author was also partially supported by the Bulgarian Ministry for Education, Science and Technology under grant MM-94$\$#415.
Article copyright: © Copyright 1996 American Mathematical Society

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