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Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Abstract

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump.

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References

  1. Agoshkov, V.I., Lebedev, V.I.: Generalized Schwarz algorithm with variable parameters. Sov. J. Numer. Anal. Math. Model. 5, 1–26 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  2. Benamou, J.-D., Després, B.: A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comp. Phys. 136, 68–82 (1997)

    MATH  Article  Google Scholar 

  3. Brenner, S.C.: The condition number of the Schur complement in domain decomposition. Numer. Math. 83, 187–203 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  4. Deng, Q.: An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput. 18, 1517–1525 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  5. Dubois, O.: Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients. Ph.D. thesis, McGill University (2007)

  6. Engquist, B., Zhao, H.-K.: Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27, 341–365 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  7. Farhat, C., Roux, F.X.: Implicit parallel processing in structural mechanics. Comput. Mech. Adv. 2, 1–124 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  8. Flauraud, E., Nataf, F., Willien, F.: Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients. J. Comput. Appl. Math. 189, 539–554 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  9. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  10. Gander, M.J., Magoules, F., Nataf, F.: Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24, 38–60 (2001)

    Article  MathSciNet  Google Scholar 

  11. Gander, M.J., Golub, G.H.: A nonoverlapping optimized Schwarz method which converges with arbitrary weak dependence on h. In: Herrera, I., Keyes, D.E., Widlund, O.B., Yates, R. (eds.) Fourteen International Conference on Domain Decomposition Methods in Science and in Engineering, pp. 281–287. DDM.org., Mexico (2003)

  12. Gander, M.J., Halpern, L., Nataf, F.: Optimized Schwarz methods. In: Chan, T., Kako, T., Kawarada, H., Pironneau, O. (eds.) Twelveth International Conference on Domain Decomposition Methods in Science and in Engineering, pp. 15–28. DDM.org., Japan (2001)

  13. Klawonn, A., Widlund, O.B., Dryja, M.: Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40, 159–179 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  14. Lions, P.L.: On the Schwarz alternating method III. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O. (eds.) In: Third Int. Symp. on Domain Decomposition Methods, pp. 202–223. SIAM, Philadelphia (1990)

  15. Lui, S.H.: A Lions nonoverlapping domain decomposition method for domains with an arbitray interface. IMA J. Numer. Anal. (2008). doi:10.1093/imanum/drm011

    Google Scholar 

  16. Maday, Y., Magoules, F.: Optimized Schwarz methods without overlap for highly heterogeneous media. Comput. Methods Appl. Mech. Eng. 196, 1541–1553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 65, 1387–1401 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  18. Nataf, F.: Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. Numer. Math. 75, 357–377 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  19. Qin, L., Xu, X.: On a parallel Robin–type nooverlapping domain decomposition method. SIAM J. Numer. Anal. 44, 2539–2558 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  20. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  21. Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschr. Nat.forsch. Ges. Zür. 15, 272–286 (1870)

    Google Scholar 

  22. Smith, B.F., Bjorstad, P., Gropp, W.D.: Domain Decomposition: Parallel Multilevel Algorithms for Elliptic Partial Differential Equations. Cambridge University Press, New York (1996)

    MATH  Google Scholar 

  23. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)

    MATH  Google Scholar 

  24. Xu, J., Zou, J.: Some nonoverlapping domain decomposition methods. SIAM Rev. 40, 857–914 (1998)

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to S. H. Lui.

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This work was in part supported by a grant from NSERC.

In memory of Gene Golub.

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Dubois, O., Lui, S.H. Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients. Numer Algor 51, 115–131 (2009). https://doi.org/10.1007/s11075-009-9268-1

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  • DOI: https://doi.org/10.1007/s11075-009-9268-1

Keywords

  • Domain decomposition
  • Convergence acceleration
  • Poincaré–Steklov operator
  • Optimized Schwarz methods
  • Discontinuous coefficient

Mathematics Subject Classifications (2000)

  • 65N55
  • 65N30
  • 65Y10
  • 35J20