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This paper develops an enhanced finite element method for approximating aclass of variational problems which exhibit the \textit{Lavrentiev gapphenomenon} in the sense that the minimum values of the energy functional havea 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 toconverge for such variational problems, a simple and effective cut-offprocedure is utilized to design the (enhanced finite element) discrete energyfunctional. In essence the proposed discrete energy functional curbs the gapphenomenon 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 theproblem-dependent parameter $\a$. Extensive 1-D and 2-D numerical experimentresults are provided to show the convergence behavior and the performance ofthe proposed enhanced finite element method.

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