An ℎ𝑝-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

In this paper we develop an h p 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 h p hp -version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust h p hp -version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.