An Enhanced Finite Element Method for a Class of Variational Problems Exhibiting the Lavrentiev Gap Phenomenon (Communications in Computational Physics, 2018)
This paper develops an enhanced finite element method for approximating a class of variational problems which exhibits the $Lavrentiev$ $gap$ $phenomenon$ in thesense that the minimum values of the energy functional have a nontrivial gap whenthe functional is minimized on the spaces $W^{1,1}$ and $W^{1,∞}$. To remedy the standard finite element method, which fails to converge for such variational problems, a simpleand effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs thegap phenomenon by capping the derivatives of its input on a scale of $\mathcal{O}$($h^{−α}$) (where $h$ denotes the mesh size) for some positive constant $α$. A sufficient condition is proposed for determining the problem-dependent parameter $α$. Extensive 1-D and 2-Dnumerical experiment results are provided to show the convergence behavior and theperformance of the proposed enhanced finite element method.
Keywords: Energy functional; variational problems; minimizers; singularities; Lavrentiev gap phenomenon; finite element methods; cut-off procedure.