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On Monotone and Schwarz Alternating Methodsfor Nonlinear Elliptic PDEs

Published online by Cambridge University Press:  15 April 2002

Shiu-Hong Lui
Shiu-Hong Lui*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. ([email protected])
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Abstract

The Schwarz alternating method can be used to solveelliptic boundary value problems on domains which consist of two or moreoverlapping subdomains. The solution is approximated by an infinite sequence offunctions 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 fornonlinear elliptic problems which are known to have solutions by the monotonemethod (also known as the method of subsolutions and supersolutions) aregiven. In particular, an additive Schwarz method for scalar as well as somecoupled nonlinear PDEs are shown to converge for finitely many subdomains.These results are applicable to several models in population biology.

Keywords

Domain decompositionnonlinear elliptic PDESchwarz alternating methodmonotone methodssubsolutionsupersolution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 1 , January 2001 , pp. 1 - 15
Copyright
© EDP Sciences, SMAI, 2001

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References

Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev. 18 (1976) 620-709. CrossRef
Badea, L., On the schwarz alternating method with more than two subdomains for nonlinear monotone problems. SIAM J. Numer. Anal. 28 (1991) 179-204. CrossRef
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.
T.F. Chan and T.P. Mathew, Domain decomposition algorithms. Acta Numer. (1994) 61-143.
M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method. BIT (1997) 296-311.
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).
R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux Eds., First Int. Symp. on Domain Decomposition Methods. SIAM, Philadelphia (1988).
Gui, C. and Lou, Y., Uniqueness and nonuniqueness of coexistence states in the lotka-volterra competition model. CPAM 47 (1994) 1571-1594.
Keller, H.B. and Cohen, D.S., Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16 (1967) 1361-1376.
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.
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.
S.H. Lui, On Schwarz alternating methods for the full potential equation. Preprint (1999).
Lui, S.H., Schwarz, On alternating methods for nonlinear elliptic pdes. SIAM J. Sci. Comput. 21 (2000) 1506-1523. CrossRef
S.H. Lui, On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. (to appear).
C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992).
Pao, C.V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations. Numer. Math. 72 (1995) 239-262. CrossRef
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999).
Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972) 979-1000. CrossRef
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).
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.
Tai, X.C. and Espedal, M., Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35 (1998) 1558-1570. CrossRef
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).
Le Tallec, P., Domain decomposition methods in computational mechanics. Computational Mechanics Advances 1 (1994) 121-220.
Two-grid di, J. Xuscretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33 (1996) 1759-1777.
Xu, J. and Zou, J., Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 857-914. CrossRef
Zou, J. and Huang, H.-C., Algebraic subproblem decomposition methods and parallel algorithms with monotone convergence. J. Comput. Math. 10 (1992) 47-59.