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Parallel two-level Schwarz methods for singularly perturbed elliptic problems

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Abstract

Schwarz domain decomposition methods are developed for the numerical solution of singularly perturbed elliptic problems. Three variants of a two-level Schwarz method with interface subproblems are investigated both theoretically and from the point of view of their computer realization on a distributed memory multiprocessor computer. Numerical experiments illustrate their parallel performance as well as their behavior with respect to the critical parameters such as the perturbation parameter, the size of the interface subdomains and the number of parallel processors. Application of one of the methods to a model problem with an interior layer of complex geometry is also discussed.

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References

  1. N. Bakhvalov, On optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969) 841–859 (in Russian).

    MATH  Google Scholar 

  2. I. Boglaev, A numerical method for a quasilinear singular perturbation problem of elliptic type, Comput. Math. Math. Phys. 28 (1988) 492–502.

    MATH  MathSciNet  Google Scholar 

  3. I. Boglaev and V. Sirotkin, Domain decomposition technique for singularly perturbed problems and its parallel implementation, in: Proc. of the 13th IMACS Congress on Computational and Applied Mathematics, eds. J. Miller and R. Vichenevetsky, Dublin (1991) pp. 522–523.

  4. I. Boglaev and V. Sirotkin, Application of domain decomposition to semiconductor device modelling and simulation in diagnostics of semiconductor structures, in: Applications of Advanced Computational Methods for Boundary and Interior Layers, ed. J. Miller (Boole, Dublin, 1993) pp. 1–32.

    Google Scholar 

  5. B. Buzbee, F. Dorr, J. George and G. Golub, The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal. 8 (1970) 722–736.

    Article  MathSciNet  Google Scholar 

  6. X.C. Cai, Multiplicative Schwarz algorithms for parabolic problems, SIAM J. Sci. Comput. 15 (1994) 587–603.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Chan and T. Mathew, Domain decomposition algorithms, Acta Numerica (1994) 61–143.

  8. R. Chin, G. Hedstrom, J. McGraw and F. Howes, Parallel computation of multiple scale problems, in: New Computing Environments: Parallel, Vector and Systolic, ed. A. Wouk (SIAM, Philadelphia, PA, 1986) pp. 136–153.

    Google Scholar 

  9. M. Garbey, A Schwarz alternating procedure for singular perturbation problems, SIAM J. Sci. Comput. 17 (1996) 1175–1201.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Hedstrom and F. Howes, A domain decomposition method for a convection diffusion equation with turning point, in: Domain Decomposition Methods, eds. T. Chan, R. Glowinski, J. Periaux and O. Widlund (SIAM, Philadelphia, PA, 1989) pp. 38–46.

    Google Scholar 

  11. Yu. Kuznetsov, Overlapping domain decomposition methods for FE-problems with elliptic singular perturbed operators, in: 4th Internat. Sympos. on Domain Decomposition Methods for Partial Differential Equations, eds. R. Glowinski, Yu. Kuznetsov, G. Meurant, J. Periaux and O. Widlund (SIAM, Philadelphia, PA, 1991) pp. 223–241.

    Google Scholar 

  12. O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations (Academic Press, New York, 1968).

    Google Scholar 

  13. P. Li and R. Peskin, Domain decomposition for singular perturbation PDEs, Math. Comput. Simulation 36 (1994) 443–455.

    Article  MathSciNet  Google Scholar 

  14. P.L. Lions, On the Schwarz alternating method I, in: 1st Internat. Sympos. on Domain Decomposition Methods for Partial Differential Equations, eds. R. Glowinski, G. Golub, G. Meurant and J. Periaux (SIAM, Philadelphia, PA, 1988) pp. 2–42.

    Google Scholar 

  15. J. Meijerink and H. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp. 31 (1977) 148–162.

    Article  MATH  MathSciNet  Google Scholar 

  16. Message Passing Interface Forum, MPI: A message-passing interface standard, Technical Report CS-94–230, Computer Science Department, University of Tennessee, Knoxville, TN (1994).

  17. J. Ortega and W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).

    Google Scholar 

  18. G. Rodrigue and E. Reiter, A domain decomposition method for boundary layer problems, in: Domain Decomposition Methods, eds. T. Chan, R. Glowinski, J. Periaux and O. Widlund (SIAM, Philadelphia, PA, 1989) pp. 226–234.

    Google Scholar 

  19. H. Schwarz, Über einige Abbildungsaufgaben, Ges. Math. Abh. 11 (1869) 65–83.

    Google Scholar 

  20. J. Scroggs, Physically motivated domain decomposition method for singularly perturbed equations, SIAM J. Numer. Anal. 28 (1991) 168–178.

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Sirotkin, The solution of singularly perturbed semilinear elliptic problem by iterative domain decomposition algorithms, Comput. Math. Appl. 33(11) (1997) 99–116.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Sirotkin, The solution of singularly perturbed elliptic problems by two-level parallel Schwarz method using ″Neumann-Dirichlet″ scheme (submitted to Appl. Numer. Math.).

  23. V. Sirotkin, Two-level parallel Schwarz methods for singularly perturbed semi-linear elliptic problems (to appear in Comput. Math. Appl.).

  24. V. Sirotkin and P. Tarvainen, Parallel numerical algorithms based on overlapping Schwarz methods for singularly perturbed semi-linear elliptic problems, Technical Report 24, University of Jyväskylä, Department of Mathematics, Laboratory of Scientific Computing (1996).

  25. V. Sirotkin and P. Tarvainen, Overlapping Schwarz methods for singularly perturbed semi-linear elliptic problems and their parallel implementation (to appear in SIAM J. Sci. Comput.).

  26. B. Smith, P. Bjørstad and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge Univ. Press, New York, 1996).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Microelectronics Technology, Russian Academy of Sciences, 142432, Chernogolovka, Moscow District, Russia

    V. Sirotkin

  2. Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35, FIN-40351, Jyväskylä, Finland E-mail: pht@mit.jyu.fi

    P. Tarvainen

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  1. V. Sirotkin
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  2. P. Tarvainen
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Sirotkin, V., Tarvainen, P. Parallel two-level Schwarz methods for singularly perturbed elliptic problems. Numerical Algorithms 22, 129–156 (1999). https://doi.org/10.1023/A:1019102723038

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