An $hp$-Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems

TL;DR

This paper introduces a fully adaptive $hp$-Newton-Galerkin method for solving semilinear elliptic boundary value problems, effectively handling singular perturbations with robust a posteriori residual analysis.

Contribution

It combines adaptive Newton methods with $hp$-finite element discretization, providing a novel, reliable scheme for complex semilinear boundary value problems.

Findings

Demonstrates robustness across various examples

Achieves reliable error control with a posteriori analysis

Effective handling of singular perturbations

Abstract

In this paper we develop an $hp$-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an $hp$-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully $hp$-adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.