$Γ$-convergence of an Enhanced Finite Element Method for Manià's Problem Exhibiting the Lavrentiev Gap Phenomenon

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 previous work by Schnake and Feng, where a promising enhanced finite element method was proposed to overcome the LGP in the classical Mani`a's problem. The 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 cruxes of the convergence analysis are taking advantage of the regularity of the minimizer and viewing the minimization problem as posed over the fractional Sobolev space $W^{1 + s, p}$ (for $s > 0$) rather than the original admissible space.