On the numerical solution of some semilinear elliptic problems.
KENDALL E. ATKINSON, Weimin Han · 2004
. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems whose leading differential operators are the Laplacian. A problem is first transformed to one on a standard domain via a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral operator equation. We employ the Galerkin method to solve the integral operator equation, using the eigenfunctions of the Laplacian on the standard domain. An error analysis of the method is given. 1 Introduction The purpose of the paper is to propose a general framework for the numerical solution of a semilinear elliptic boundary value problem of the form 8 ! : \\Gamma\\DeltaU = F (\\Delta; U) in\\Omega ; U = G on @\\Omega ; (1:1) where\\Omega ae R 2 is a simply-connected domain with a boundary @ For the domain\\Omega\\Gamma assume there exists a standard domain D, such that\\Omega and D are conformally equivalent. The first step of the method is to re...