Numerical analysis of semilinear elliptic equations with finite spectral interaction

TL;DR

This paper introduces an algorithm for solving certain semilinear elliptic equations with finite spectral interaction, extending previous geometric methods and addressing non-resonant nonlinearities with finite eigenvalue interactions.

Contribution

It develops a novel algorithm for semilinear elliptic equations with finite spectral interaction, building on geometric proof techniques and previous work.

Findings

Algorithm successfully solves equations with finite spectral interaction.

Extends geometric proof methods to broader class of nonlinearities.

Addresses non-resonant cases with finite eigenvalue spectra.

Abstract

We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichlet boundary conditions in a bounded domain $\Omega$. The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$ is an endpoint of $\bar{\partial_2f(\Omega,\RR)}$, which in turn only contains a finite number of eigenvalues. The algorithm is based in ideas used by Berger and Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem and advances work by Smiley and Chun for the same problem.