Fully Adaptive Newton-Galerkin Methods for Semilinear Elliptic Partial Differential Equations

TL;DR

This paper introduces a fully adaptive Newton-Galerkin method for solving semilinear elliptic PDEs, combining adaptive Newton iterations with finite element discretization, validated through numerical experiments demonstrating robustness.

Contribution

It presents a novel fully adaptive scheme integrating Newton's method with finite element discretization for semilinear elliptic problems, including singular perturbations.

Findings

Robustness demonstrated across various examples

Effective a posteriori error control

Reliable convergence of the adaptive scheme

Abstract

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.