---
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.'
address:
- 'Lucerne University of Applied Sciences and Arts, CH-6002 Luzern, Switzerland'
- ' Institut für Analysis und Scientific Computing, TU Wien, A-1040 Wien, Austria'
- 'Mathematics Institute, University of Bern, CH-3012 Switzerland'
author:
- Mario Amrein
- 'Jens M. Melenk'
- 'Thomas P. Wihler'
bibliography:
- 'references.bib'
title: 'An $hp$-Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems'
---

Introduction
============

The purpose of this paper is the development of a numerical approximation procedure for semilinear elliptic boundary value problems, with possible singular perturbations. More precisely, for a fixed parameter ${\epsilon}>0$ (possibly with ${\epsilon}\ll 1$), and a continuously differentiable function $f:\,\Omega\times\mathbb{R}\to\mathbb{R}$, we consider the problem of finding a function $u:\,\Omega\to\mathbb{R}$ in an interval $\Omega=(a,b)$, $a<b$, which satisfies $$\label{eq:poisson}
\begin{aligned}
-{\epsilon}u''(x) &=f(x,u(x)) \text{ for }x\in \Omega,\qquad
