Weak Galerkin method for the Biot’s consolidation model

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

doi:10.1016/j.camwa.2017.07.013

Title: Weak Galerkin method for the Biot’s consolidation model

Abstract

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.

Authors:: Hu, Xiaozhe; Mu, Lin; Ye, Xiu

Publication Date:: Thu Mar 01 00:00:00 EST 2018

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

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1854979

Alternate Identifier(s):: OSTI ID: 1394349; OSTI ID: 1582708

Grant/Contract Number:: ERKJE45; AC05-00OR22725

Resource Type:: Published Article

Journal Name:: Computers and Mathematics with Applications (Oxford)

Additional Journal Information:: Journal Name: Computers and Mathematics with Applications (Oxford) Journal Volume: 75 Journal Issue: 6; Journal ID: ISSN 0898-1221

Publisher:: Elsevier

Country of Publication:: United Kingdom

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Biot’s consolidation model; Finite element method; Weak Galerkin finite elements method

Citation Formats


                    Hu, Xiaozhe, Mu, Lin, and Ye, Xiu. Weak Galerkin method for the Biot’s consolidation model.  United Kingdom: N. p., 2018. 
Web.  doi:10.1016/j.camwa.2017.07.013.

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                    Hu, Xiaozhe, Mu, Lin, & Ye, Xiu. Weak Galerkin method for the Biot’s consolidation model.  United Kingdom.  https://doi.org/10.1016/j.camwa.2017.07.013

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                    Hu, Xiaozhe, Mu, Lin, and Ye, Xiu. Thu .  
"Weak Galerkin method for the Biot’s consolidation model".  United Kingdom.  https://doi.org/10.1016/j.camwa.2017.07.013.

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@article{osti_1854979,

  title        = {Weak Galerkin method for the Biot’s consolidation model},

  author       = {Hu, Xiaozhe and Mu, Lin and Ye, Xiu},

  abstractNote = {In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.},

  doi          = {10.1016/j.camwa.2017.07.013},

  journal      = {Computers and Mathematics with Applications (Oxford)},

  number       = 6,

  volume       = 75,

  place        = {United Kingdom},

  year         = {Thu Mar 01 00:00:00 EST 2018},

  month        = {Thu Mar 01 00:00:00 EST 2018}

}

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Title: Weak Galerkin method for the Biot’s consolidation model

Abstract

Citation Formats

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A nonconforming finite element method for the Biot’s consolidation model in poroelasticity journal, January 2017

Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium journal, June 1999

A stable finite element for the stokes equations journal, December 1984

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case journal, March 2007

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Coupling versus uncoupling in soil consolidation journal, August 1991

On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems journal, January 2007

Stabilized Lowest-Order Finite Element Approximation for Linear Three-Field Poroelasticity journal, January 2015

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Mandel's problem revisited journal, June 1996

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Asymptotic Behavior of Semidiscrete Finite-Element Approximations of Biot’s Consolidation Problem journal, June 1996

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Numerical simulation of secondary consolidation of soil: Finite element application journal, January 1989

On stability and convergence of finite element approximations of Biot's consolidation problem journal, February 1994

Finite Difference Schemes for Poro-elastic ProblemS journal, January 2002

On the causes of pressure oscillations in low-permeable and low-compressible porous media: PRESSURE OSCILLATIONS IN LOW-PERMEABLE POROUS MEDIA journal, July 2011

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model journal, February 2013

A modified weak Galerkin finite element method for the Stokes equations journal, February 2015

Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach journal, December 2008

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

Improved accuracy in finite element analysis of Biot's consolidation problem journal, March 1992

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation
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A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods
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A nonconforming finite element method for the Biot’s consolidation model in poroelasticity
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Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium
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A stable finite element for the stokes equations
journal, December 1984

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case
journal, March 2007

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case
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Coupling versus uncoupling in soil consolidation
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On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems
journal, January 2007

Stabilized Lowest-Order Finite Element Approximation for Linear Three-Field Poroelasticity
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Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity
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A weak Galerkin finite element method for second-order elliptic problems
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A weak Galerkin mixed finite element method for second order elliptic problems
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A computational study of the weak Galerkin method for second-order elliptic equations
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Mandel's problem revisited
journal, June 1996

A fully coupled consolidation model of the subsidence of Venice
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Convergence analysis of a new mixed finite element method for Biot's consolidation model
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General Theory of Three‐Dimensional Consolidation
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A fully coupled 3-D mixed finite element model of Biot consolidation
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Diffusion in Poro-Elastic Media
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A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation
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Asymptotic Behavior of Semidiscrete Finite-Element Approximations of Biot’s Consolidation Problem
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A finite difference analysis of Biot's consolidation model
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Consolidation Des Sols (Étude Mathématique)
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Stability and monotonicity for some discretizations of the Biot’s consolidation model
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Theory of Elasticity and Consolidation for a Porous Anisotropic Solid
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The Pore-Pressure Coefficients A and B
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Numerical simulation of secondary consolidation of soil: Finite element application
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On stability and convergence of finite element approximations of Biot's consolidation problem
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