Weak Galerkin methods for second order elliptic interface problems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:250

Article

Mu, Lin¹  Wang, Junping²  Wei, Guowei¹  Ye, Xiu³  Zhao, Shan⁴ 

[1] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

[2] Natl Sci Fdn, Div Math Sci, Arlington, VA 22230 USA

[3] Univ Arkansas, Dept Math & Stat, Little Rock, AR 72204 USA

[4] Univ Alabama, Dept Math, Tuscaloosa, AL 35487 USA

关键词: Finite element methods;    Weak Galerkin method;    Second order elliptic interface problems;    Nonsmooth interface;    Low solution regularity;   

DOI  :  10.1016/j.jcp.2013.04.042

来源: Elsevier

【 摘 要 】

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 out 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-infinity 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 O(h) to O(h(1.5)) for the solution itself in L-infinity 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 O(h(1.75)) to O(h(2)) in the L-infinity norm for C-1 or Lipschitz continuous interfaces associated with a C-1 or H-2 continuous solution. (C) 2013 Elsevier Inc. All rights reserved.

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