On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs

Shiu-Hong Lui

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 1, page 1-15
  • ISSN: 0764-583X

Abstract

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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 of the subdomains. 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 as some coupled nonlinear PDEs are shown to converge for finitely many subdomains. These results are applicable to several models in population biology.

How to cite

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Lui, Shiu-Hong. "On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs." ESAIM: Mathematical Modelling and Numerical Analysis 35.1 (2010): 1-15. <http://eudml.org/doc/197538>.

@article{Lui2010,
abstract = { 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 of the subdomains. 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 as some coupled nonlinear PDEs are shown to converge for finitely many subdomains. These results are applicable to several models in population biology. },
author = {Lui, Shiu-Hong},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Domain decomposition; nonlinear elliptic PDE; Schwarz alternating method; monotone methods; subsolution; supersolution.; domain decomposition; Schwarz alternating method; nonlinear elliptic PDEs; convergence; difference methods; population biology},
language = {eng},
month = {3},
number = {1},
pages = {1-15},
publisher = {EDP Sciences},
title = {On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs},
url = {http://eudml.org/doc/197538},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Lui, Shiu-Hong
TI - On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 1
SP - 1
EP - 15
AB - 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 of the subdomains. 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 as some coupled nonlinear PDEs are shown to converge for finitely many subdomains. These results are applicable to several models in population biology.
LA - eng
KW - Domain decomposition; nonlinear elliptic PDE; Schwarz alternating method; monotone methods; subsolution; supersolution.; domain decomposition; Schwarz alternating method; nonlinear elliptic PDEs; convergence; difference methods; population biology
UR - http://eudml.org/doc/197538
ER -

References

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  1. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev.18 (1976) 620-709.  
  2. L. Badea, On the schwarz alternating method with more than two subdomains for nonlinear monotone problems. SIAM J. Numer. Anal.28 (1991) 179-204.  
  3. X.C. Cai and M. Dryja, Domain decomposition methods for monotone nonlinear elliptic problems, in Domain decomposition methods in scientific and engineering computing, D. Keyes and J. Xu Eds., AMS, Providence, R.I. (1994) 335-360.  
  4. T.F. Chan and T.P. Mathew, Domain decomposition algorithms. Acta Numer. (1994) 61-143.  
  5. M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method. BIT (1997) 296-311.  
  6. 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, New York, USA (1987).  
  7. R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux Eds., First Int. Symp. on Domain Decomposition Methods. SIAM, Philadelphia (1988).  
  8. C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the lotka-volterra competition model. CPAM47 (1994) 1571-1594.  
  9. H.B. Keller and D.S. Cohen, Some positone problems suggested by nonlinear heat generation. J. Math. Mech.16 (1967) 1361-1376.  
  10. P.L. Lions, On the Schwarz alternating method I, in First Int. Symp. on Domain Decomposition Methods, R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux Eds., SIAM, Philadelphia (1988) 1-42.  
  11. P.L. Lions, On the Schwarz alternating method II, in Second Int. Conference on Domain Decomposition Methods, T.F. Chan, R. Glowinski, J. Periaux and O. Widlund Eds., SIAM, Philadelphia (1989) 47-70.  
  12. S.H. Lui, On Schwarz alternating methods for the full potential equation. Preprint (1999).  
  13. S.H. Lui, On Schwarz alternating methods for nonlinear elliptic pdes. SIAM J. Sci. Comput.21 (2000) 1506-1523.  
  14. S.H. Lui, On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. (to appear).  
  15. C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992).  
  16. C.V. Pao, Block monotone iterative methods for numerical solutions of nonlinear elliptic equations. Numer. Math.72 (1995) 239-262.  
  17. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999).  
  18. D.H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J.21 (1972) 979-1000.  
  19. B.F. Smith, P. Bjorstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Algorithms for Elliptic Partial Differential Equations. Cambridge University Press, New York (1996).  
  20. X.C. Tai, Domain decomposition for linear and nonlinear elliptic problems via function or space decomposition, in Domain decomposition methods in scientific and engineering computing, D. Keyes and J. Xu Eds., AMS, Providence, R.I. (1994) 335-360.  
  21. X.C. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal.35 (1998) 1558-1570.  
  22. X.C. Tai and J. Xu, Global convergence of subspace correction methods for convex optimization problems. Report 114, Department of Mathematics, University of Bergen, Norway (1998).  
  23. P. Le Tallec, Domain decomposition methods in computational mechanics. Computational Mechanics Advances1 (1994) 121-220.  
  24. J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.33 (1996) 1759-1777.  
  25. J. Xu and J. Zou, Some nonoverlapping domain decomposition methods. SIAM Rev.40 (1998) 857-914.  
  26. J. Zou and H.-C. Huang, Algebraic subproblem decomposition methods and parallel algorithms with monotone convergence. J. Comput. Math.10 (1992) 47-59.  

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