 
Summary: JOURNAL OF INTEGRAL EQUATIONS
AND APPLICATIONS
Volume 17, Number 3, Fall 2005
SOLVING THE NONLINEAR POISSON EQUATION
ON THE UNIT DISK
KENDALL ATKINSON AND OLAF HANSEN
ABSTRACT. We propose and analyze a numerical method
for solving the nonlinear Poisson equation u = f(·, u) on
the unit disk with zero Dirichlet boundary conditions. The
problem is reformulated as a nonlinear integral equation. We
use a Galerkin method with polynomials as approximations.
The speed of convergence is shown to be very rapid; and ex
perimentally the maximum error is exponentially decreasing
when it is regarded as a function of the degree of the approx
imating polynomial.
1. Introduction. In the earlier papers [2, 4] a Galerkin method
was proposed, analyzed, and illustrated for the numerical solution of a
Dirichlet problem for a semilinear elliptic boundary value problem of
the form
(1)
