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On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

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Summary

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain.

In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology.

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References

  1. H. Amann: Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev.18, 620–709 (1976)

    Article  MATH  Google Scholar 

  2. L. Badea: On the schwarz alternating method with more than two subdomains for nonlinear monotone problems. SIAM J. Numer. Anal.28, 179–204 (1991)

    Article  MATH  Google Scholar 

  3. X. C. Cai, M. Dryja: Domain decomposition methods for monotone nonlinear elliptic problems. In D. Keyes, J. Xu, editors, Domain decomposition methods in scientific and engineering computing, pp. 335–360, Providence, R.I., 1994. AMS

  4. X. C. Cai, D. E. Keyes: Nonlinearly preconditioned inexact newton algorithms. SIAM J. Sci. Comput. (to appear)

  5. T. F. Chan, T. P. Mathew: Domain decomposition algorithms. Acta Numerica, pp. 61–143, 1994

  6. G. Chen, Y. Deng, W. M. Ni, J. Zhou: Boundary element monotone iteration scheme for semilinear elliptic partial differential equations. Math. Comp.65, 943–982 (1996)

    Article  MATH  Google Scholar 

  7. G. Chen, Y. Deng, W. M. Ni, J. Zhou: Boundary element monotone iteration scheme for semilinear elliptic partial differential equations part ii. Math. Comp.69, 629–652 (2000)

    Article  MATH  Google Scholar 

  8. M. Dryja, W. Hackbusch: On the nonlinear domain decomposition method. BIT, pp. 296–311, 1997

  9. M. Dryja, O. B. Widlund: An additive variant of the Schwarz alternating method for the case of many subregions. Technical report, Courant Institute Technical Report 339, 1987

  10. R. Glowinski, G. H. Golub, G. A. Meurant, J. Periaux: First Int. Symp. on Domain Decomposition Methods. SIAM, Philadelphia, 1988

    Google Scholar 

  11. C. Gui, Y. Lou: Uniqueness and nonuniqueness of coexistence states in the lotkavolterra competition model. CPAM47, 1571–1594 (1994)

    MATH  Google Scholar 

  12. D. K. Kaushik, D. E. Keyes, B. F. Smith: On the interaction of architecture and algorithm in the domain-based parallelization of an unstructured-grid incompressible flow code. In J. Mandel, C. Farhat, X. C. Cai, editors, Domain Decomposition Methods 10, pp. 287–295, Providence, 1998. AMS

    Google Scholar 

  13. H.B. Keller, D.S. Cohen: Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16(12):1361–1376, 1967

    MATH  Google Scholar 

  14. P. L. Lions: On the Schwarz alternating method I. In R. Glowinski, G. H. Golub, G. A. Meurant, J. Periaux, editors, First Int. Symp. on Domain Decomposition Methods, pp. 1–42, Philadelphia, 1988. SIAM

    Google Scholar 

  15. P. L. Lions: On the Schwarz alternating method II. In T. F. Chan, R. Glowinski, J. Periaux, O. Widlund, editors, Second Int. Conference on Domain Decomposition Methods, pp. 47–70, Philadelphia, 1989. SIAM

    Google Scholar 

  16. S. H. Lui: On Schwarz alternating methods for nonlinear elliptic pdes. SIAM J. Sci. Comput.21, 1506–1523 (2000)

    Article  MATH  Google Scholar 

  17. S. H. Lui: On monotone and Schwarz alternating methods for nonlinear elliptic PDEs. M2AN35, 1–15 (2001)

    Article  MATH  Google Scholar 

  18. S. H. Lui: On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput.22, 1974–1986 (2001)

    Article  MATH  Google Scholar 

  19. C. V. Pao: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, 1992

    MATH  Google Scholar 

  20. C. V. Pao: Block monotone iterative methods for numerical solutions of nonlinear elliptic equations. Numer. Math.72, 239–262 (1995)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. D. H. Sattinger: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J.21, 979–1000 (1972)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. X. C. Tai: Domain decomposition for linear and nonlinear elliptic problems via function or space decomposition. In D. Keyes, J. Xu, editors, Domain decomposition methods in scientific and engineering computing, pp. 335–360, Providence, R.I., 1994. AMS

    Google Scholar 

  25. X. C. Tai, M. Espedal: Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal.35, 1558–1570 (1998)

    Article  MATH  Google Scholar 

  26. X. C. Tai, J. Xu: Global convergence of subspace correction methods for convex optimization problems. Math. Comp. (to appear)

  27. P. Le Tallec: Domain decomposition methods in computational mechanics. Computational Mechanics Advances1, 121–220 (1994)

    MATH  Google Scholar 

  28. J. Xu: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.33, 1759–1777 (1996)

    Article  MATH  Google Scholar 

  29. J. Zou, H.-C. Huang: Algebriac subproblem decomposition methods and parallel algorithms with monotone convergence. J. Comput. Math.10, 47–59 (1992)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Clear Water Bay, Hong Kong University of Science & Technology, Kowloon, Hong Kong

    S. H. Lui

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  1. S. H. Lui
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This work was in part supported by a grant from the RGC of HKSAR, China (HKUST6171/99P)

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Lui, S.H. On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math. 93, 109–129 (2002). https://doi.org/10.1007/BF02679439

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