An enhanced finite element method for a class of variational problems exhibiting the Lavrentiev gap phenomenon

TL;DR

This paper introduces an enhanced finite element method with a cut-off procedure to effectively approximate variational problems exhibiting the Lavrentiev gap phenomenon, ensuring convergence where standard methods fail.

Contribution

The paper proposes a novel enhanced finite element approach that incorporates a cut-off to handle the Lavrentiev gap, improving convergence for challenging variational problems.

Findings

Method successfully handles Lavrentiev gap in 1-D and 2-D problems.

Enhanced method demonstrates convergence where standard FEM does not.

Numerical results confirm the effectiveness of the cut-off procedure.

Abstract

This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the \textit{Lavrentiev gap phenomenon} in the sense that the minimum values of the energy functional have a nontrivial gap when the functional is minimized on spaces $W^{1,1}$ and $W^{1,\infty}$. To remedy the standard finite element method, which fails to converge for such variational problems, a simple and effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs the gap phenomenon by capping the derivatives of its input on a scale of $O(h^{-\alpha})$ (where $h$ denotes the mesh size) for some positive constant $\alpha$. A sufficient condition is proposed for determining the problem-dependent parameter $\a$. Extensive 1-D and 2-D numerical experiment results are provided to show the convergence behavior and the performance of the proposed enhanced finite element method.