Adaptive Energy Minimisation for $hp$-Finite Element Methods

Abstract: This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of $hp$-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new $hp$-refinement technique for both one- and two-dimensional problems.