TY - JOUR

T1 - Boundary element monotone iteration scheme for semilinear elliptic partial differential equations, part II

T2 - quasimonotone iteration for coupled 2 × 2 systems

AU - Chen, Goong

AU - Deng, Yuanhua

AU - Ni, Wei Ming

AU - Zhou, Jianxin

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2000/4

Y1 - 2000/4

N2 - Numerical solutions of 2 × 2 sernilinear 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 × 2 matrix problem, the norm of which must be estimated. Prom the analysis of such 2 × 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.

AB - Numerical solutions of 2 × 2 sernilinear 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 × 2 matrix problem, the norm of which must be estimated. Prom the analysis of such 2 × 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.

KW - Boundary elements

KW - Error analysis

KW - Lotka-Volterra models

KW - Nonlinear elliptic systems

UR - http://www.scopus.com/inward/record.url?scp=0034420434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034420434&partnerID=8YFLogxK

U2 - 10.1090/s0025-5718-99-01109-6

DO - 10.1090/s0025-5718-99-01109-6

M3 - Article

AN - SCOPUS:0034420434

VL - 69

SP - 629

EP - 652

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 230

ER -