A weak Galerkin finite element method with polynomial reduction

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:285

Article

Mu, Lin¹  Wang, Junping²  Ye, Xiu³ 

[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, Little Rock, AR 72204 USA

关键词: Weak Galerkin;    Finite element methods;    Weak gradient;    Discrete weak gradient;    Second-order elliptic equation;    Polyhedral meshes;   

DOI  :  10.1016/j.cam.2015.02.001

来源: Elsevier

【 摘 要 】

The weak Galerkin (WG) is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme is significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction. Consequently, a new weak Galerkin finite element method is proposed and analyzed. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H-1 norm and the standard L-2 norm. In addition, the paper presents some numerical results to demonstrate the power of the WG method in dealing with finite element partitions with arbitrary polygons in 2D or polyhedra in 3D. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honeycomb mesh in 2D and mesh with deformed cubes in 3D. (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

Free