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

Abstract

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.

Authors:
 [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:
Research Org.:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1329764
Alternate Identifier(s):
OSTI ID: 1459032
Grant/Contract Number:  
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 325; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Finite element method; Weak Galerkin method; Elliptic interface problem; Nonsmooth interface; Low solution regularity; High order method

Citation Formats

Mu, Lin, Wang, Junping, Ye, Xiu, and Zhao, Shan. A new weak Galerkin finite element method for elliptic interface problems. United States: N. p., 2016. Web. doi:10.1016/j.jcp.2016.08.024.
Mu, Lin, Wang, Junping, Ye, Xiu, & Zhao, Shan. A new weak Galerkin finite element method for elliptic interface problems. United States. https://doi.org/10.1016/j.jcp.2016.08.024
Mu, Lin, Wang, Junping, Ye, Xiu, and Zhao, Shan. 2016. "A new weak Galerkin finite element method for elliptic interface problems". United States. https://doi.org/10.1016/j.jcp.2016.08.024. https://www.osti.gov/servlets/purl/1329764.
@article{osti_1329764,
title = {A new weak Galerkin finite element method for elliptic interface problems},
author = {Mu, Lin and Wang, Junping and Ye, Xiu and Zhao, Shan},
abstractNote = {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.},
doi = {10.1016/j.jcp.2016.08.024},
url = {https://www.osti.gov/biblio/1329764}, journal = {Journal of Computational Physics},
issn = {0021-9991},
number = C,
volume = 325,
place = {United States},
year = {Fri Aug 26 00:00:00 EDT 2016},
month = {Fri Aug 26 00:00:00 EDT 2016}
}

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Cited by: 76 works
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Works referencing / citing this record:

A least-squares-based weak Galerkin finite element method for elliptic interface problems
journal, July 2019


Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods
journal, November 2019


Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition
journal, January 2020


Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions
journal, November 2019


A partially penalty immersed Crouzeix-Raviart finite element method for interface problems
journal, August 2017