A new weak Galerkin finite element method for elliptic interface problems

Mu, Lin; Wang, Junping; Ye, Xiu; Zhao, Shan

doi:10.1016/j.jcp.2016.08.024

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

Journal Article · Fri Aug 26 00:00:00 EDT 2016 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2016.08.024· OSTI ID:1329764

Mu, Lin ^[1]; Wang, Junping ^[2]; Ye, Xiu ^[3];

^[4]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
National Science Foundation, Arlington, VA (United States). Division of Mathematical Sciences
Univ. of Arkansas, Little Rock, AR (United Sates). Dept. of Mathematics and Statistics
Univ. of Alabama, Tuscaloosa, AL (United States). Dept. of Mathematics

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.

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Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1329764

Alternate ID(s):: OSTI ID: 1459032

Journal Information:: Journal of Computational Physics, Vol. 325, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 76 works

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References (53)

The finite element method for elliptic equations with discontinuous coefficients Babuška, Ivo Computing, Vol. 5, Issue 3 https://doi.org/10.1007/BF02248021	journal	September 1970
A second order virtual node method for elliptic problems with interfaces and irregular domains Bedrossian, Jacob; von Brecht, James H.; Zhu, Siwei Journal of Computational Physics, Vol. 229, Issue 18 https://doi.org/10.1016/j.jcp.2010.05.002	journal	September 2010
A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions Berthelsen, Petter Andreas Journal of Computational Physics, Vol. 197, Issue 1 https://doi.org/10.1016/j.jcp.2003.12.003	journal	June 2004
A finite element method for interface problems in domains with smooth boundaries and interfaces Bramble, James H.; King, J. Thomas Advances in Computational Mathematics, Vol. 6, Issue 1 https://doi.org/10.1007/BF02127700	journal	December 1996
Two families of mixed finite elements for second order elliptic problems Brezzi, Franco; Douglas, Jim; Marini, L. D. Numerische Mathematik, Vol. 47, Issue 2 https://doi.org/10.1007/BF01389710	journal	June 1985
Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems Burman, E.; Hansbo, P. IMA Journal of Numerical Analysis, Vol. 30, Issue 3 https://doi.org/10.1093/imanum/drn081	journal	April 2009
Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations Cai, Zhiqiang; Ye, Xiu; Zhang, Shun SIAM Journal on Numerical Analysis, Vol. 49, Issue 5 https://doi.org/10.1137/100805133	journal	January 2011
MIBPB: A software package for electrostatic analysis Chen, Duan; Chen, Zhan; Chen, Changjun Journal of Computational Chemistry, Vol. 32, Issue 4 https://doi.org/10.1002/jcc.21646	journal	September 2010
Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems Chen, Tianbing; Strain, John Journal of Computational Physics, Vol. 227, Issue 16 https://doi.org/10.1016/j.jcp.2008.04.027	journal	August 2008
Finite element methods and their convergence for elliptic and parabolic interface problems Chen, Zhiming; Zou, Jun Numerische Mathematik, Vol. 79, Issue 2 https://doi.org/10.1007/s002110050336	journal	April 1998
A coupling interface method for elliptic interface problems Chern, I-Liang; Shu, Yu-Chen Journal of Computational Physics, Vol. 225, Issue 2 https://doi.org/10.1016/j.jcp.2007.03.012	journal	August 2007
BDDC methods for discontinuous Galerkin discretization of elliptic problems Dryja, Maksymilian; Galvis, Juan; Sarkis, Marcus Journal of Complexity, Vol. 23, Issue 4-6 https://doi.org/10.1016/j.jco.2007.02.003	journal	August 2007
The immersed finite volume element methods for the elliptic interface problems Ewing, Richard E.; Li, Zhilin; Lin, Tao Mathematics and Computers in Simulation, Vol. 50, Issue 1-4 https://doi.org/10.1016/S0378-4754(99)00061-0	journal	November 1999
A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) Fedkiw, Ronald P.; Aslam, Tariq; Merriman, Barry Journal of Computational Physics, Vol. 152, Issue 2 https://doi.org/10.1006/jcph.1999.6236	journal	July 1999
Treatment of charge singularities in implicit solvent models Geng, Weihua; Yu, Sining; Wei, Guowei The Journal of Chemical Physics, Vol. 127, Issue 11 https://doi.org/10.1063/1.2768064	journal	September 2007
A fictitious domain method for Dirichlet problem and applications Glowinski, Roland; Pan, Tsorng-Whay; Periaux, Jacques Computer Methods in Applied Mechanics and Engineering, Vol. 111, Issue 3-4 https://doi.org/10.1016/0045-7825(94)90135-X	journal	January 1994
A discontinuous Galerkin method for elliptic interface problems with application to electroporation Guyomarc'h, Grégory; Lee, Chang-Ock; Jeon, Kiwan Communications in Numerical Methods in Engineering, Vol. 25, Issue 10 https://doi.org/10.1002/cnm.1132	journal	October 2009
High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces Hadley, G. R. Journal of Lightwave Technology, Vol. 20, Issue 7 https://doi.org/10.1109/JLT.2002.800361	journal	July 2002
An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems Hansbo, Anita; Hansbo, Peter Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 47-48 https://doi.org/10.1016/S0045-7825(02)00524-8	journal	November 2002
Analysis of an efficient finite element method for embedded interface problems Harari, Isaac; Dolbow, John Computational Mechanics, Vol. 46, Issue 1 https://doi.org/10.1007/s00466-009-0457-5	journal	December 2009
Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient He, Xiaoming; Lin, Tao; Lin, Yanping Journal of Systems Science and Complexity, Vol. 23, Issue 3 https://doi.org/10.1007/s11424-010-0141-z	journal	June 2010
A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions Hellrung, Jeffrey Lee; Wang, Luming; Sifakis, Eftychios Journal of Computational Physics, Vol. 231, Issue 4 https://doi.org/10.1016/j.jcp.2011.11.023	journal	February 2012
Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems Hiptmair, R.; Li, J.; Zou, J. Journal of Numerical Mathematics, Vol. 18, Issue 3 https://doi.org/10.1515/jnum.2010.010	journal	January 2010
Modal analysis of circular Bragg fibers with arbitrary index profiles Horikis, Theodoros P.; Kath, William L. Optics Letters, Vol. 31, Issue 23 https://doi.org/10.1364/OL.31.003417	journal	January 2006
A numerical method for solving variable coefficient elliptic equation with interfaces Hou, Songming; Liu, Xu-Dong Journal of Computational Physics, Vol. 202, Issue 2, p. 411-445 https://doi.org/10.1016/j.jcp.2004.07.016	journal	January 2005
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces Hou, Songming; Wang, Wei; Wang, Liqun Journal of Computational Physics, Vol. 229, Issue 19 https://doi.org/10.1016/j.jcp.2010.06.005	journal	September 2010
A weak formulation for solving elliptic interface problems without body fitted grid Hou, Songming; Song, Peng; Wang, Liqun Journal of Computational Physics, Vol. 249 https://doi.org/10.1016/j.jcp.2013.04.025	journal	September 2013
A Hybrid Method for Moving Interface Problems with Application to the Hele–Shaw Flow Hou, Thomas Y.; Li, Zhilin; Osher, Stanley Journal of Computational Physics, Vol. 134, Issue 2 https://doi.org/10.1006/jcph.1997.5689	journal	July 1997
A high-order hybridizable discontinuous Galerkin method for elliptic interface problems: HDG METHOD FOR ELLIPTIC INTERFACE PROBLEMS Huynh, L. N. T.; Nguyen, N. C.; Peraire, J. International Journal for Numerical Methods in Engineering, Vol. 93, Issue 2 https://doi.org/10.1002/nme.4382	journal	June 2012
Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces Layton, Anita T. Computers & Fluids, Vol. 38, Issue 2 https://doi.org/10.1016/j.compfluid.2008.02.003	journal	February 2009
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources LeVeque, Randall J.; Li, Zhilin SIAM Journal on Numerical Analysis, Vol. 31, Issue 4, p. 1019-1044 https://doi.org/10.1137/0731054	journal	August 1994
Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients Li, Zhilin; Ito, Kazufumi SIAM Journal on Scientific Computing, Vol. 23, Issue 1 https://doi.org/10.1137/S1064827500370160	journal	January 2001
An Unfitted Discontinuous Galerkin Method Applied to Elliptic Interface Problems Massjung, Ralf SIAM Journal on Numerical Analysis, Vol. 50, Issue 6 https://doi.org/10.1137/090763093	journal	January 2012
The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions Mayo, Anita SIAM Journal on Numerical Analysis, Vol. 21, Issue 2 https://doi.org/10.1137/0721021	journal	April 1984
Weak Galerkin methods for second order elliptic interface problems Mu, Lin; Wang, Junping; Wei, Guowei Journal of Computational Physics, Vol. 250 https://doi.org/10.1016/j.jcp.2013.04.042	journal	October 2013
A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces Oevermann, M.; Klein, R. Journal of Computational Physics, Vol. 219, Issue 2 https://doi.org/10.1016/j.jcp.2006.04.010	journal	December 2006
A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid Peskin, Charles S.; McQueen, David M. Journal of Computational Physics, Vol. 81, Issue 2 https://doi.org/10.1016/0021-9991(89)90213-1	journal	April 1989
Numerical analysis of blood flow in the heart Peskin, Charles S. Journal of Computational Physics, Vol. 25, Issue 3 https://doi.org/10.1016/0021-9991(77)90100-0	journal	November 1977
Convergence analysis of the Q ₁ -finite element method for elliptic problems with non-boundary-fitted meshes Ramière, Isabelle International Journal for Numerical Methods in Engineering, Vol. 75, Issue 9 https://doi.org/10.1002/nme.2278	journal	August 2008
An Edge-based Anisotropic Mesh Refinement Algorithm and Its Application to Interface Problems Wang, Duan Communications in Computational Physics, Vol. 8, Issue 3 https://doi.org/10.4208/cicp.210709.121109a	journal	June 2010
A weak Galerkin finite element method for second-order elliptic problems Wang, Junping; Ye, Xiu Journal of Computational and Applied Mathematics, Vol. 241 https://doi.org/10.1016/j.cam.2012.10.003	journal	March 2013
A weak Galerkin mixed finite element method for second order elliptic problems Wang, Junping; Ye, Xiu Mathematics of Computation, Vol. 83, Issue 289 https://doi.org/10.1090/S0025-5718-2014-02852-4	journal	May 2014
On computational issues of immersed finite element methods Sheldon Wang, X.; Zhang, L. T.; Liu, Wing Kam Journal of Computational Physics, Vol. 228, Issue 7 https://doi.org/10.1016/j.jcp.2008.12.012	journal	April 2009
The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions Wiegmann, Andreas; Bube, Kenneth P. SIAM Journal on Numerical Analysis, Vol. 37, Issue 3 https://doi.org/10.1137/S0036142997328664	journal	January 2000
MIB Galerkin method for elliptic interface problems Xia, Kelin; Zhan, Meng; Wei, Guo-Wei Journal of Computational and Applied Mathematics, Vol. 272 https://doi.org/10.1016/j.cam.2014.05.014	journal	December 2014
A Galerkin formulation of the MIB method for three dimensional elliptic interface problems Xia, Kelin; Wei, Guo-Wei Computers & Mathematics with Applications, Vol. 68, Issue 7 https://doi.org/10.1016/j.camwa.2014.07.022	journal	October 2014
A kernel-free boundary integral method for implicitly defined surfaces Ying, Wenjun; Wang, Wei-Cheng Journal of Computational Physics, Vol. 252 https://doi.org/10.1016/j.jcp.2013.06.019	journal	November 2013
Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities Yu, Sining; Wei, G. W. Journal of Computational Physics, Vol. 227, Issue 1 https://doi.org/10.1016/j.jcp.2007.08.003	journal	November 2007
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces Yu, Sining; Zhou, Yongcheng; Wei, G. W. Journal of Computational Physics, Vol. 224, Issue 2 https://doi.org/10.1016/j.jcp.2006.10.030	journal	June 2007
Treatment of geometric singularities in implicit solvent models Yu, Sining; Geng, Weihua; Wei, G. W. The Journal of Chemical Physics, Vol. 126, Issue 24 https://doi.org/10.1063/1.2743020	journal	June 2007
High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces Zhao, Shan Journal of Computational Physics, Vol. 229, Issue 9 https://doi.org/10.1016/j.jcp.2009.12.034	journal	May 2010
High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces Zhao, Shan; Wei, G. W. Journal of Computational Physics, Vol. 200, Issue 1 https://doi.org/10.1016/j.jcp.2004.03.008	journal	October 2004
High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources Zhou, Y. C.; Zhao, Shan; Feig, Michael Journal of Computational Physics, Vol. 213, Issue 1 https://doi.org/10.1016/j.jcp.2005.07.022	journal	March 2006

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71 CLASSICAL AND QUANTUM MECHANICS
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Title: A new weak Galerkin finite element method for elliptic interface problems

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