MATHEMATICS OF COMPUTATION Volume 65, Number 215 July 1996, Pages 943–982 BOUNDARY ELEMENT MONOTONE ITERATION SCHEME FOR SEMILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS YUANHUA DENG, GOONG CHEN, WEI-MING NI, AND JIANXIN ZHOU Abstract. The monotone iteration scheme is a constructive method for solv- ing a wide class of semilinear elliptic boundary value problems. With the avail- ability of a supersolution and a subsolution, the iterates converge monotoni- cally 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 δ for the numerical implementation of boundary elements within the range of monotone convergence. We then in- terrelate 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 (Ω), 0 ≤ r ≤ 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 sta- bility are discussed, computed, and the graphics of their numerical solutions are illustrated. 1. Introduction Numerical solutions of nonlinear partial differential equations (PDEs) are im- portant in applications. Historically, such work is done primarily by the finite difference methods (FDM) and finite element methods (FEM). While boundary element methods (BEM) have steadily gained popularity among engineers and sci- entists in their work of computing solutions of PDEs, owing to the very nature of their formulation, BEM are still regarded by many people as mainly applicable to Received by the editor May 4, 1994 and, in revised form, March 1, 1995. 1991 Mathematics Subject Classification. Primary 31B20, 35J65, 65N38. Key words and phrases. Numerical PDE, boundary elements, potential theory, nonlinear PDE, elliptic type. 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. c 1996 American Mathematical Society 943 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use