An $hp$-Adaptive Newton-Discontinuous-Galerkin Finite Element Approach for Semilinear Elliptic Boundary Value Problems

TL;DR

This paper introduces an $hp$-adaptive Newton-DG finite element method for solving second-order semilinear elliptic boundary value problems, effectively handling singular perturbations with proven robustness and reliability.

Contribution

It combines adaptive Newton schemes with an $hp$-discontinuous Galerkin discretization based on a posteriori residual analysis, advancing numerical solutions for complex elliptic problems.

Findings

Demonstrates robustness across various test cases

Achieves reliable error control with $hp$-adaptivity

Effective handling of singular perturbations

Abstract

In this paper we develop an $hp$-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an $hp$-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust $hp$-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.