An Enhanced Finite Element Method for a Class of Variational Problems Exhibiting the Lavrentiev Gap Phenomenon
Year: 2018
Communications in Computational Physics, Vol. 24 (2018), Iss. 2 : pp. 576–592
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2017-0046
Communications in Computational Physics, Vol. 24 (2018), Iss. 2 : pp. 576–592
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: Energy functional variational problems minimizers singularities Lavrentiev gap phenomenon finite element methods cut-off procedure.