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Mathematics of Computation

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Boundary element monotone iteration scheme
for semilinear elliptic
partial differential equations,
Part II: Quasimonotone iteration
for coupled $2\times 2$ systems

Authors: Goong Chen, Yuanhua Deng, Wei-Ming Ni and Jianxin Zhou
Journal: Math. Comp. 69 (2000), 629-652
MSC (1991): Primary 31B20, 35J65, 65N30
Published electronically: August 24, 1999
MathSciNet review: 1651745
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Abstract | References | Similar Articles | Additional Information

Abstract: Numerical solutions of $2\times 2$ semilinear systems of elliptic boundary value problems, whose nonlinearities are of quasimonotone nondecreasing, quasimonotone nonincreasing, or mixed quasimonotone types, are computed. At each step of the (quasi) monotone iteration, the solution is represented by a simple-layer potential plus a domain integral; the simple-layer density is then discretized by boundary elements. Because of the various combinations of Dirichlet, Neumann and Robin boundary conditions, there is an associated $2\times 2$ matrix problem, the norm of which must be estimated. From the analysis of such $2\times 2$ matrices, we formulate conditions which guarantee the monotone iteration a strict contraction staying within the close range of a given pair of subsolution and supersolution. Thereafter, boundary element error analysis can be carried out in a similar way as for the discretized problem. A concrete example of a monotone dissipative system on a 2D annular domain is also computed and illustrated.

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Additional Information

Goong Chen
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Yuanhua Deng
Affiliation: Northern Telecom, 2201 Lakeside Blvd, Richardson, Texas 75082

Wei-Ming Ni
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Jianxin Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Keywords: Nonlinear elliptic systems, boundary elements, error analysis, Lotka-Volterra models
Received by editor(s): July 15, 1996
Received by editor(s) in revised form: April 30, 1998
Published electronically: August 24, 1999
Additional Notes: Professors Chen and Zhou were supported in part by NSF Grants DMS 9404380 and 9610076
Professor Ni was supported in part by NSF Grants DMS 9401333 and 9705639
Article copyright: © Copyright 2000 American Mathematical Society