Rapidly Convergent Approximations to Dirichlet's Problem for Semilinear Elliptic Equations

Abstract

In this paper an iterative scheme is given for solving the Dirichlet problem for the semilinear elliptic equation, ▵u = f(x, y, u, u_x u_y), defined in a simply connected domain with Lyapunov boundary, The procedure developed makes use of the kernel function methods of Bergman and Schiffer to estimate Green's function for the equation ▵u-α²u. Using this Green's function the Dirichlet problem is then reformulated as an integral equation. It is shown that the approximants converge to the actual solution geometrically. The convergence factor is the reciprocal of the first eigenvalue to a particular Fredholm integral equation. This eigenvalue is known to be always greater than 1