Solving the Nonlinear Poisson Equation on the Unit Disk

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 experimentally the maximum error is exponentially decreasing when it is regarded as a function of the degree of the approximating polynomial.