TY - JOUR
T1 - An Enhanced Finite Element Method for a Class of Variational Problems Exhibiting the Lavrentiev Gap Phenomenon
AU - Xiaobing Feng & Stefan Schnake
JO - Communications in Computational Physics
VL - 2
SP - 576
EP - 592
PY - 2018
DA - 2018/08
SN - 24
DO - http://doi.org/10.4208/cicp.OA-2017-0046
UR - https://global-sci.org/intro/article_detail/cicp/12253.html
KW - Energy functional, variational problems, minimizers, singularities, Lavrentiev gap phenomenon, finite element methods, cut-off procedure.
AB - <p style="text-align: justify;">This paper develops an enhanced finite element method for approximating a class of variational problems which exhibits the $Lavrentiev$ $gap$ $phenomenon$ in the
sense that the minimum values of the energy functional have a nontrivial gap when
the 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 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 $\mathcal{O}$($h^{−α}$) (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-D
numerical experiment results are provided to show the convergence behavior and the
performance of the proposed enhanced finite element method.</p>