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Domain decomposition algorithms for mixed
methods for second-order elliptic problems


Authors: Zhangxin Chen, Richard E. Ewing and Raytcho Lazarov
Journal: Math. Comp. 65 (1996), 467-490
MSC (1991): Primary {65N30, 65N22, 65F10}
DOI: https://doi.org/10.1090/S0025-5718-96-00703-X
MathSciNet review: 1333307
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Abstract: In this paper domain decomposition algorithms for mixed finite element methods for linear second-order elliptic problems in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ are developed. A convergence theory for two-level and multilevel Schwarz methods applied to the algorithms under consideration is given. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques.


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  • 1. T. Arbogast and Zhangxin Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), 943--972. CMP 95:04
  • 2. D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), 7--32. MR 87g:65126
  • 3. J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), 1--21. MR 92d:65094
  • 4. ------, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 57 (1991), 23--46. MR 91m:65158
  • 5. S. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Preprint.
  • 6. F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), 237--250. MR 88f:65190
  • 7. F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), 581--604. MR 88j:65249
  • 8. F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217--235. MR 87g:65133
  • 9. Zhangxin Chen, On the existence, uniqueness and convergence of nonlinear mixed finite element methods, Mat. Apl. Comput. 8 (1989), 241--258. MR 91g:65139
  • 10. ------, Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 27 (1993), 9--34. MR 94c:65132
  • 11. ------, BDM mixed methods for a nonlinear elliptic problem, J. Comp. Appl. Math. 53 (1994), 207--223. CMP 95:05
  • 12. ------, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems, East-West J. Numer. Math. 4 (1996) (to appear).
  • 13. Zhangxin Chen and J. Douglas, Jr., Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems, Mat. Apl. Comput. 10 (1991), 137--160. MR 93d:65097
  • 14. ------, Prismatic mixed finite elements for second order elliptic problems, Calcolo 26 (1989), 135--148. MR 92e:65148
  • 15. P. Ciarlet, The Finite Element Method for Elliptic Problems, North--Holland, Amsterdam, 1978. MR 58:25001
  • 16. L. Cowsar, Dual-variable Schwarz methods for mixed finite elements, Dept. Comp. and Appl. Math. TR 93-09, Rice University, 1993.
  • 17. ------, Domain decomposition methods for nonconforming finite element spaces of Lagrange type, in the Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, N. Melson et al., eds., NASA Conference Publication 3224 Part 1 (1993), 93--109.
  • 18. L. Cowsar, J. Mandel, and M. Wheeler, Balancing domain decomposition for mixed finite elements, Dept. Comp. and Appl. Math. TR 93-08, Rice University, 1993.
  • 19. L. Cowsar and M. Wheeler, Parallel domain decomposition method for mixed finite elements for elliptic partial differential equations, Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al., eds., SIAM, 1991. CMP 91:12
  • 20. Y. De Roeck and P. Le Tallec, Analysis and test of a local domain decomposition preconditioner, Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al., eds., SIAM, 1991. CMP 91:12
  • 21. J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math. 65 (1993), 95--108. MR 94c:65134
  • 22. J. Douglas, Jr. and J. Wang, A new family of mixed finite element spaces over rectangles, Mat. Apl. Comput. 12 (1993), 183--197. CMP 94:16
  • 23. M. Dryja and O. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Statist. Comput. 15 (1994), 604--620. MR 95d:65102
  • 24. R. Ewing, R. Lazarov, T. Russell, and P. Vassilevski, Local refinement via domain decomposition techniques for mixed finite element methods withrectangular Raviart-Thomas elements, Proc. Third Int. Symp. on DD Methods for PDE's, T. Chan et al., eds., SIAM, Philadelphia, 1990, pp. 98--114. MR 91f:65191
  • 25. R. Ewing and J. Wang, Analysis of the Schwarz algorithm for mixed finite element methods, RAIRO Modél. Math. Anal. Numér. 26 (1992), 739--756. MR 94c:65135
  • 26. ------, Analysis of multilevel decomposition iterative methods for mixed finite element methods, RAIRO Modél. Math. Anal. Numér 28 (1994), 377--398. MR 95e:65099
  • 27. R. Glowinski, W. Kinton, and M. Wheeler, Acceleration of domain decomposition algorithms for mixed finite elements by multilevel methods, Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al., eds., SIAM, 1990, pp. 263--290. CMP 90:15
  • 28. R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al., eds., SIAM, 1988, pp. 144--172. MR 90a:65237
  • 29. J. Mandel, Balancing domain decomposition, Comm. Numer. Methods Engrg. 9 (1993), 233--241. MR 94b:65158
  • 30. J. Mandel and M. Brezina, Balancing domain decomposition: theory and performance in two and three dimensions, to appear.
  • 31. T. P. Mathew, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: algorithms and numerical results, Numer. Math. 65 (1993), 445--468. MR 94m:65171
  • 32. ------, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II: convergence theory, Numer. Math. 65 (1993), 469--492. MR 94m:65172
  • 33. F. Milner, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp. 44 (1985), 303--320. MR 86g:65215
  • 34. J. C. Nedelec, Mixed finite elements in $\text{\bf R}^{3}$, Numer. Math. 35 (1980), 315--341. MR 81k:65125
  • 35. ------, A new family of mixed finite elements in $\text{\bf R}^{3}$, Numer. Math. 50 (1986), 57--81. MR 88e:65145
  • 36. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977, pp. 292--315. MR 58:3547
  • 37. M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, N. Melson et al., eds., NASA Conference Publication 3224 Part 2 (1993), 543--566.
  • 38. B. Smith, An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems, SIAM J. Sci. Statist. Comput. 13 (1992), 364--378.
  • 39. P. 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
  • 40. ------, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), 503--520. MR 93j:65187

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Additional Information

Zhangxin Chen
Affiliation: Department of Mathematics and the Institute for Scientific Computation, Texas A$&$M University, College Station, TX 77843
Address at time of publication: Department of Mathematics, Box 156, Southern Methodist University, Dallas, Texas 75275-0156
Email: zchen@isc.tamu.edu

Richard E. Ewing
Affiliation: Department of Mathematics and the Institute for Scientific Computation, Texas A$&$M University, College Station, TX 77843
Email: ewing@ewing.tamu.edu

Raytcho Lazarov
Affiliation: Department of Mathematics and the Institute for Scientific Computation, Texas A$&$M University, College Station, TX 77843
Email: lazarov@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-96-00703-X
Keywords: Finite element, implementation, mixed method, conforming and nonconforming methods, domain decomposition, convergence, projection of coefficient
Received by editor(s): August 2, 1994
Received by editor(s) in revised form: March 21, 1995
Additional Notes: Partly supported by the Department of Energy under contract DE-ACOS-840R21400.
Article copyright: © Copyright 1996 American Mathematical Society

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