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

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:

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

Publication Date:: Fri Aug 26 00:00:00 EDT 2016

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:: 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.

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                    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

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                    Mu, Lin, Wang, Junping, Ye, Xiu, and Zhao, Shan. Fri .  
"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.

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@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},

  journal      = {Journal of Computational Physics},

  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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Works referencing / citing this record:

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journal, July 2019

Deka, Bhupen; Roy, Papri
Proceedings - Mathematical Sciences, Vol. 129, Issue 5
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Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods
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He, Cuiyu; Zhang, Xu
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Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition
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Zhou, Chenguang; Zou, Yongkui; Chai, Shimin
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Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions
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Deka, Bhupen; Roy, Papri
Numerical Methods for Partial Differential Equations, Vol. 36, Issue 4
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A partially penalty immersed Crouzeix-Raviart finite element method for interface problems
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An, Na; Yu, Xijun; Chen, Huanzhen
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DOI: 10.1186/s13660-017-1461-5

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

Abstract

Citation Formats

The finite element method for elliptic equations with discontinuous coefficients journal, September 1970

A second order virtual node method for elliptic problems with interfaces and irregular domains journal, September 2010

A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions journal, June 2004

A finite element method for interface problems in domains with smooth boundaries and interfaces journal, December 1996

Two families of mixed finite elements for second order elliptic problems journal, June 1985

Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems journal, April 2009

Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations journal, January 2011

MIBPB: A software package for electrostatic analysis journal, September 2010

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems journal, August 2008

Finite element methods and their convergence for elliptic and parabolic interface problems journal, April 1998

A coupling interface method for elliptic interface problems journal, August 2007

BDDC methods for discontinuous Galerkin discretization of elliptic problems journal, August 2007

The immersed finite volume element methods for the elliptic interface problems journal, November 1999

A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) journal, July 1999

Treatment of charge singularities in implicit solvent models journal, September 2007

A fictitious domain method for Dirichlet problem and applications journal, January 1994

A discontinuous Galerkin method for elliptic interface problems with application to electroporation journal, October 2009

High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces journal, July 2002

An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems journal, November 2002

Analysis of an efficient finite element method for embedded interface problems journal, December 2009

Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient journal, June 2010

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions journal, February 2012

Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems journal, January 2010

Modal analysis of circular Bragg fibers with arbitrary index profiles journal, January 2006

A numerical method for solving variable coefficient elliptic equation with interfaces journal, January 2005

Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces journal, September 2010

A weak formulation for solving elliptic interface problems without body fitted grid journal, September 2013

A Hybrid Method for Moving Interface Problems with Application to the Hele–Shaw Flow journal, July 1997

A high-order hybridizable discontinuous Galerkin method for elliptic interface problems: HDG METHOD FOR ELLIPTIC INTERFACE PROBLEMS journal, June 2012

Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces journal, February 2009

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources journal, August 1994

Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients journal, January 2001

An Unfitted Discontinuous Galerkin Method Applied to Elliptic Interface Problems journal, January 2012

The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions journal, April 1984

Weak Galerkin methods for second order elliptic interface problems journal, October 2013

A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal, December 2006

A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid journal, April 1989

Numerical analysis of blood flow in the heart journal, November 1977

Convergence analysis of the Q 1 -finite element method for elliptic problems with non-boundary-fitted meshes journal, August 2008

An Edge-based Anisotropic Mesh Refinement Algorithm and Its Application to Interface Problems journal, June 2010

A weak Galerkin finite element method for second-order elliptic problems journal, March 2013

A weak Galerkin mixed finite element method for second order elliptic problems journal, May 2014

On computational issues of immersed finite element methods journal, April 2009

The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions journal, January 2000

MIB Galerkin method for elliptic interface problems journal, December 2014

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems journal, October 2014

A kernel-free boundary integral method for implicitly defined surfaces journal, November 2013

Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities journal, November 2007

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces journal, June 2007

Treatment of geometric singularities in implicit solvent models journal, June 2007

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces journal, May 2010

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces journal, October 2004

High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources journal, March 2006

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

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The finite element method for elliptic equations with discontinuous coefficients
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A second order virtual node method for elliptic problems with interfaces and irregular domains
journal, September 2010

A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions
journal, June 2004

A finite element method for interface problems in domains with smooth boundaries and interfaces
journal, December 1996

Two families of mixed finite elements for second order elliptic problems
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Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems
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Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations
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MIBPB: A software package for electrostatic analysis
journal, September 2010

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems
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Finite element methods and their convergence for elliptic and parabolic interface problems
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A coupling interface method for elliptic interface problems
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The immersed finite volume element methods for the elliptic interface problems
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A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method)
journal, July 1999

Treatment of charge singularities in implicit solvent models
journal, September 2007

A fictitious domain method for Dirichlet problem and applications
journal, January 1994

A discontinuous Galerkin method for elliptic interface problems with application to electroporation
journal, October 2009

High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces
journal, July 2002

An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems
journal, November 2002

Analysis of an efficient finite element method for embedded interface problems
journal, December 2009

Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient
journal, June 2010

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions
journal, February 2012

Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems
journal, January 2010

Modal analysis of circular Bragg fibers with arbitrary index profiles
journal, January 2006

A numerical method for solving variable coefficient elliptic equation with interfaces
journal, January 2005

Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
journal, September 2010

A weak formulation for solving elliptic interface problems without body fitted grid
journal, September 2013

A Hybrid Method for Moving Interface Problems with Application to the Hele–Shaw Flow
journal, July 1997

A high-order hybridizable discontinuous Galerkin method for elliptic interface problems: HDG METHOD FOR ELLIPTIC INTERFACE PROBLEMS
journal, June 2012

Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces
journal, February 2009

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
journal, August 1994

Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients
journal, January 2001

An Unfitted Discontinuous Galerkin Method Applied to Elliptic Interface Problems
journal, January 2012

The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions
journal, April 1984

Weak Galerkin methods for second order elliptic interface problems
journal, October 2013

A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces
journal, December 2006

A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid
journal, April 1989

Numerical analysis of blood flow in the heart
journal, November 1977

Convergence analysis of the Q ₁ -finite element method for elliptic problems with non-boundary-fitted meshes
journal, August 2008

An Edge-based Anisotropic Mesh Refinement Algorithm and Its Application to Interface Problems
journal, June 2010

A weak Galerkin finite element method for second-order elliptic problems
journal, March 2013

A weak Galerkin mixed finite element method for second order elliptic problems
journal, May 2014

On computational issues of immersed finite element methods
journal, April 2009

The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions
journal, January 2000

MIB Galerkin method for elliptic interface problems
journal, December 2014

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems
journal, October 2014

A kernel-free boundary integral method for implicitly defined surfaces
journal, November 2013

Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities
journal, November 2007

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
journal, June 2007

Treatment of geometric singularities in implicit solvent models
journal, June 2007

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces
journal, May 2010

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces
journal, October 2004

High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources
journal, March 2006

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