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Boundary element monotone iteration
scheme for semilinear elliptic
partial differential equations

Authors: Yuanhua Deng, Goong Chen, Wei-Ming Ni and Jianxin Zhou
Journal: Math. Comp. 65 (1996), 943-982
MSC (1991): Primary 31B20, 35J65, 65N38
MathSciNet review: 1348042
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Abstract: The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin $\delta $ for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the $H^{r}(\Omega )$, $0\le r \le 2$, Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, ``higher than optimal order'' error estimates can be obtained with respect to the mesh parameter $h$. Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

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Additional Information

Yuanhua Deng
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

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

Wei-Ming Ni
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Jianxin Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843; School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Keywords: Numerical PDE, boundary elements, potential theory, nonlinear PDE, elliptic type
Received by editor(s): May 4, 1994
Received by editor(s) in revised form: March 1, 1995
Additional Notes: The first, second, and fourth authors were supported in part by AFOSR Grant 91-0097 and NSF Grant DMS 9404380.
The third author was supported in part by NSF Grants DMS 9101446 and 9401333.
The second author was on sabbatical leave at the Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan, ROC. Supported in part by a grant from the National Science Council of the Republic of China.
Article copyright: © Copyright 1996 American Mathematical Society