Adaptive Energy Minimisation for $hp$-Finite Element Methods

TL;DR

This paper introduces an adaptive $hp$-finite element method that iteratively refines the approximation space to efficiently solve convex variational problems, demonstrating promising numerical results in 1D and 2D cases.

Contribution

It presents a novel $hp$-refinement technique for finite element methods, enabling adaptive enrichment of the approximation space for better solution accuracy.

Findings

Effective $hp$-refinement for linear and nonlinear problems

Improved numerical performance in 1D and 2D cases

Demonstrated convergence and efficiency of the method

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.