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WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS.
MedLine Citation:
PMID:  24072935     Owner:  NLM     Status:  Publisher    
Abstract/OtherAbstract:
Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L 2 and L ∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order [Formula: see text] to [Formula: see text] for the solution itself in L ∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order [Formula: see text] to [Formula: see text] in the L ∞ norm for C (1) or Lipschitz continuous interfaces associated with a C (1) or H (2) continuous solution.
Authors:
Lin Mu; Junping Wang; Guowei Wei; Xiu Ye; Shan Zhao
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Publication Detail:
Type:  JOURNAL ARTICLE    
Journal Detail:
Title:  Journal of computational physics     Volume:  250     ISSN:  0021-9991     ISO Abbreviation:  J Comput Phys     Publication Date:  2013 Oct 
Date Detail:
Created Date:  2013-9-27     Completed Date:  -     Revised Date:  -    
Medline Journal Info:
Nlm Unique ID:  9883524     Medline TA:  J Comput Phys     Country:  -    
Other Details:
Languages:  ENG     Pagination:  106-125     Citation Subset:  -    
Affiliation:
Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR 72204 ( lxmu@ualr.edu ).
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