A Numerical Algorithm for Ambrosetti-Prodi Type Operators

TL;DR

This paper presents a numerical algorithm combining finite element methods with a Lyapunov-Schmidt decomposition to solve Ambrosetti-Prodi type nonlinear elliptic equations with Dirichlet boundary conditions.

Contribution

It introduces a novel numerical approach for solving nonlinear elliptic PDEs of Ambrosetti-Prodi type using a global Lyapunov-Schmidt decomposition.

Findings

Effective numerical solution for Ambrosetti-Prodi equations

Algorithm handles nonlinearities with bounded derivatives

Potential for accurate solutions in bounded domains

Abstract

We consider the numerical solution of the equation - \Delta u - f(u) = g, for the unknown u satisfying Dirichlet conditions in a bounded domain. The nonlinearity f has bounded, continuous derivative. The algorithm uses the finite element method combined with a global Lyapunov-Schmidt decomposition.