  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.
