$\Gamma$-convergence of an Enhanced Finite Element Method for Mani\`a's and Foss's Problems Exhibiting the Lavrentiev Gap Phenomenon

TL;DR

This paper proves the $ ext{Gamma}$-convergence of an enhanced finite element method designed to overcome the Lavrentiev Gap Phenomenon in variational problems, extending its application from Manià's to Foss's problem.

Contribution

It provides a complete $ ext{Gamma}$-convergence proof for the enhanced finite element method and extends its application to two-dimensional Foss's problem.

Findings

The enhanced finite element method converges for Manià's problem.

The method is extended and shown to converge for Foss's problem.

The convergence analysis leverages fractional Sobolev space regularity.

Abstract

It is well-known that numerically approximating calculus of variations problems possessing a Lavrentiev Gap Phenomenon (LGP) is challenging, and the standard numerical methodologies, such as finite element, finite difference, and discontinuous Galerkin methods, fail to give convergent methods because they cannot overcome the gap. This paper is a continuation of a 2016 paper by Feng and Schnake, where a promising enhanced finite element method was proposed to overcome the LGP in the classical Mani\`a's problem. The first goal of this paper is to provide a complete $\Gamma$-convergence proof for this enhanced finite element method, hence, establishing a theoretical foundation for the method. The crux of the convergence analysis is taking advantage of the regularity of the minimizer and viewing the minimization problem as posed over the fractional Sobolev space $W^{1 + s, p}(0, 1)$ (for $s > 0$) rather than the original admissible space $W^{1, p}(0, 1)$. The second goal is to extend the enhanced finite element method to the two-dimensional Foss's problem from nonlinear elasticity, which is also known to possess the LGP, and to establish its $\Gamma$-convergence as well.