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Title: A new weak Galerkin finite element method for elliptic interface problems

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments in order to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.
 [1] ;  [2] ;  [3] ; ORCiD logo [4]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
  2. National Science Foundation, Arlington, VA (United States). Division of Mathematical Sciences
  3. Univ. of Arkansas, Little Rock, AR (United Sates). Dept. of Mathematics and Statistics
  4. Univ. of Alabama, Tuscaloosa, AL (United States). Dept. of Mathematics
Publication Date:
OSTI Identifier:
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 325; Journal Issue: C; Journal ID: ISSN 0021-9991
Research Org:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
Country of Publication:
United States
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Finite element method; Weak Galerkin method; Elliptic interface problem; Nonsmooth interface; Low solution regularity; High order method