A multigrid waveform relaxation method for solving the poroelasticity equations

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based o...

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Veröffentlicht in:	Computational & applied mathematics Jg. 37; H. 4; S. 4805 - 4820
Hauptverfasser:	Franco, S. R., Rodrigo, C., Gaspar, F. J., Pinto, M. A. V.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.09.2018 Springer Nature B.V
Schlagworte:	Algebra Applications of Mathematics Applied physics Computational mathematics Computational Mathematics and Numerical Analysis Finite difference method Iterative methods Mathematical Applications in Computer Science Mathematical Applications in the Physical Sciences Mathematical models Mathematics Mathematics and Statistics Poroelasticity Relaxation method (mathematics) Multigrid waveform relaxation 65F10 Poroelasticity Semi-algebraic mode analysis Space–time grids 65M55 65M22 Vanka smoother
ISSN:	0101-8205, 2238-3603, 1807-0302
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Abstract	In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.
AbstractList	In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.
Author	Rodrigo, C. Pinto, M. A. V. Franco, S. R. Gaspar, F. J.
Author_xml	– sequence: 1 givenname: S. R. surname: Franco fullname: Franco, S. R. organization: Department of Mathematics, State University of Centro-Oeste, Federal University of Paraná, Graduate Program in Numerical Methods in Engineering – sequence: 2 givenname: C. orcidid: 0000-0002-1598-2831 surname: Rodrigo fullname: Rodrigo, C. email: carmenr@unizar.es organization: IUMA and Applied Mathematics Department, University of Zaragoza – sequence: 3 givenname: F. J. surname: Gaspar fullname: Gaspar, F. J. organization: CWI, Centrum Wiskunde & Informatica – sequence: 4 givenname: M. A. V. surname: Pinto fullname: Pinto, M. A. V. organization: Department of Mechanical Engineering, Federal University of Paraná
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Cites_doi	10.1002/9780470172766 10.1137/0731087 10.1007/s10596-008-9114-x 10.1006/jcph.1997.5854 10.1016/0021-9991(86)90008-2 10.1007/BF01934186 10.1007/s00791-007-0061-1 10.1007/978-3-0348-5712-3_23 10.1090/S0025-5718-1977-0431719-X 10.1002/nla.1977 10.1002/nag.1062 10.1016/j.apnum.2016.02.006 10.1016/j.jcp.2010.03.018 10.1016/j.cma.2015.09.019 10.1137/16M1090193 10.1063/1.1721956 10.1002/nme.2295 10.1016/S0168-9274(02)00190-3 10.1137/15M1014280 10.1016/j.apnum.2007.11.014 10.1007/978-3-322-94761-1 10.1061/9780784412992.110 10.1063/1.1712886 10.1002/nla.762
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Keywords	Multigrid waveform relaxation 65F10 Poroelasticity Semi-algebraic mode analysis Space–time grids 65M55 65M22 Vanka smoother
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References	LubichCOstermannAMultigrid dynamic iteration for parabolic equationsBIT19872721623489412410.1007/BF019341860623.65125 WienandsRJoppichWPractical Fourier analysis for multigrid methods2005Boca RatonChapman and Hall/CRC Press1062.65133 Hu X, Rodrigo C, Gaspar FJ (2017) Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation. arXiv:1706.07632v1 [math.NA] WesselingPAn introduction to multigrid methods1992ChichesterWiley0760.65092 BiotMAGeneral theory of three-dimensional consolidationJ Appl Phys194112215516410.1063/1.171288667.0837.01 VandewalleSParallel Multigrid waveform relaxation for parabolic problems1993Stuttgart, B.GTeubner10.1007/978-3-322-94761-10816.65057 Favino M, Grillo A, Krause R (2013) A stability condition for the numerical simulation of poroelastic systems. In: Hellmich C, Pichler B, Adam D (eds) Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, pp 919–928. https://doi.org/10.1061/9780784412992.110 GasparFJLisbonaFJOosterleeCWA stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methodsComput Vis Sci20081126776238983610.1007/s00791-007-0061-1 BrandtARigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L2-normSIAM J Numer Anal199431616951730130268110.1137/07310870817.65126 BrandtAMultigrid solvers for non-elliptic and singular-perturbation steady-state problems1981RehovotThe Weizmann Institute of Science VankaSBlock-implicit multigrid solution of Navier-Stokes equations in primitive variablesJ Comput Phys198665113815884845110.1016/0021-9991(86)90008-20606.76035 Gaspar FJ, Rodrigo C (2017) Multigrid waveform relaxation for the time-fractional heat equation. SIAM J Sci Comput 39(4):A1201–A1224. https://doi.org/10.1137/16M1090193 Aguilar G, Gaspar F, Lisbona F, Rodrigo C (2008) Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int J Numer Meth Eng 75(11):1282–1300. https://doi.org/10.1002/nme.2295 Molenaar J (1991) A two-grid analysis of the combination of mixed finite elements and Vanka-type relaxation. In: Hackbusch W, Trottenberg U (eds) Multigrid Methods III. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 98. Birkhäuser, Basel LewisRSchreflerBThe finite element method in the static and dynamic deformation and consolidation of porous media1998New YorkWiley0935.74004 Gaspar FJ, Lisbona FJ, Vabishchevich PN (2003) A finite difference analysis of Biot’s consolidation model. Appl Numer Math 44(4):487–506. https://doi.org/10.1016/S0168-9274(02)00190-3 PhillipsPWheelerMOvercoming the problem of locking in linear elasticity and poroelasticity: an heuristic approachComput Geosci200913151210.1007/s10596-008-9114-x1172.74017 Haga JB, Osnes H, Langtangen HP (2012) On the causes of pressure oscillations in low-permeable and low-compressible porous media. Int J Numer Anal Meth Geomech 36(12):1507–1522. https://doi.org/10.1002/nag.1062 OosterleeCGasparFWashioTWienandsRMultigrid line smoothers for higher order upwind discretizations of convection-dominated problemsJ Comput Phys19981392274307161409010.1006/jcph.1997.58540908.65111 Nordbotten JM (2016) Stable cell-centered finite volume discretization for biot equations. SIAM J Numer Anal 54(2):942–968. https://doi.org/10.1137/15M1014280 OosterleeCWGasparFJMultigrid relaxation methods for systems of saddle point typeAppl Numer Math2008581219331950246482210.1016/j.apnum.2007.11.0141148.76040 FerronatoMCastellettoNGambolatiGA fully coupled 3-d mixed finite element model of Biot consolidationJ Comput Phys2010229124813483010.1016/j.jcp.2010.03.0181305.76055 RodrigoCGasparFHuXZikatanovLStability and monotonicity for some discretizations of the Biot’s consolidation modelComput Methods Appl Mech Eng2016298183204342771110.1016/j.cma.2015.09.019 TerzaghiKTheoretical soil mechanics1943New YorkWiley10.1002/9780470172766 TrottenbergUOosterleeCWSchüllerAMultigrid2001New YorkAcademic Press0976.65106 BiotMATheory of elasticity and consolidation for a porous anisotropic solidJ Appl Phys19552621821856687410.1063/1.17219560067.23603 MacLachlanSPOosterleeCWLocal fourier analysis for multigrid with overlapping smoothers applied to systems of pdesNumer Linear Algebra Appl2011184751774284019810.1002/nla.7621265.65256 BrandtAMulti-level adaptive solutions to boundary-value problemsMath Comput19773113833339043171910.1090/S0025-5718-1977-0431719-X0373.65054 GanderMJ50 years of time parallel time integration2015ChamSpringer International Publishing691131337.65127 FriedhoffSMacLachlanSA generalized predictive analysis tool for multigrid methodsNumer Linear Algebra Appl2015224618647336782610.1002/nla.19771349.65427 RodrigoCGasparFLisbonaFOn a local fourier analysis for overlapping block smoothers on triangular gridsAppl Numer Math201610596111348807610.1016/j.apnum.2016.02.0061382.65419 FJ Gaspar (603_CR13) 2008; 11 S Vanka (603_CR29) 1986; 65 C Lubich (603_CR17) 1987; 27 P Phillips (603_CR23) 2009; 13 MJ Gander (603_CR10) 2015 R Wienands (603_CR31) 2005 M Ferronato (603_CR8) 2010; 229 603_CR19 603_CR14 603_CR15 603_CR7 603_CR11 U Trottenberg (603_CR27) 2001 603_CR12 S Friedhoff (603_CR9) 2015; 22 K Terzaghi (603_CR26) 1943 C Rodrigo (603_CR25) 2016; 105 MA Biot (603_CR2) 1941; 12 603_CR1 MA Biot (603_CR3) 1955; 26 P Wesseling (603_CR30) 1992 R Lewis (603_CR16) 1998 SP MacLachlan (603_CR18) 2011; 18 S Vandewalle (603_CR28) 1993 A Brandt (603_CR6) 1994; 31 C Rodrigo (603_CR24) 2016; 298 A Brandt (603_CR5) 1981 CW Oosterlee (603_CR22) 2008; 58 A Brandt (603_CR4) 1977; 31 C Oosterlee (603_CR21) 1998; 139 603_CR20
References_xml	– reference: Gaspar FJ, Rodrigo C (2017) Multigrid waveform relaxation for the time-fractional heat equation. SIAM J Sci Comput 39(4):A1201–A1224. https://doi.org/10.1137/16M1090193 – reference: FerronatoMCastellettoNGambolatiGA fully coupled 3-d mixed finite element model of Biot consolidationJ Comput Phys2010229124813483010.1016/j.jcp.2010.03.0181305.76055 – reference: LewisRSchreflerBThe finite element method in the static and dynamic deformation and consolidation of porous media1998New YorkWiley0935.74004 – reference: BrandtARigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L2-normSIAM J Numer Anal199431616951730130268110.1137/07310870817.65126 – reference: OosterleeCWGasparFJMultigrid relaxation methods for systems of saddle point typeAppl Numer Math2008581219331950246482210.1016/j.apnum.2007.11.0141148.76040 – reference: GasparFJLisbonaFJOosterleeCWA stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methodsComput Vis Sci20081126776238983610.1007/s00791-007-0061-1 – reference: BiotMAGeneral theory of three-dimensional consolidationJ Appl Phys194112215516410.1063/1.171288667.0837.01 – reference: RodrigoCGasparFLisbonaFOn a local fourier analysis for overlapping block smoothers on triangular gridsAppl Numer Math201610596111348807610.1016/j.apnum.2016.02.0061382.65419 – reference: Hu X, Rodrigo C, Gaspar FJ (2017) Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation. arXiv:1706.07632v1 [math.NA] – reference: TrottenbergUOosterleeCWSchüllerAMultigrid2001New YorkAcademic Press0976.65106 – reference: GanderMJ50 years of time parallel time integration2015ChamSpringer International Publishing691131337.65127 – reference: Gaspar FJ, Lisbona FJ, Vabishchevich PN (2003) A finite difference analysis of Biot’s consolidation model. Appl Numer Math 44(4):487–506. https://doi.org/10.1016/S0168-9274(02)00190-3 – reference: Haga JB, Osnes H, Langtangen HP (2012) On the causes of pressure oscillations in low-permeable and low-compressible porous media. Int J Numer Anal Meth Geomech 36(12):1507–1522. https://doi.org/10.1002/nag.1062 – reference: TerzaghiKTheoretical soil mechanics1943New YorkWiley10.1002/9780470172766 – reference: OosterleeCGasparFWashioTWienandsRMultigrid line smoothers for higher order upwind discretizations of convection-dominated problemsJ Comput Phys19981392274307161409010.1006/jcph.1997.58540908.65111 – reference: BrandtAMulti-level adaptive solutions to boundary-value problemsMath Comput19773113833339043171910.1090/S0025-5718-1977-0431719-X0373.65054 – reference: Favino M, Grillo A, Krause R (2013) A stability condition for the numerical simulation of poroelastic systems. In: Hellmich C, Pichler B, Adam D (eds) Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, pp 919–928. https://doi.org/10.1061/9780784412992.110 – reference: Molenaar J (1991) A two-grid analysis of the combination of mixed finite elements and Vanka-type relaxation. In: Hackbusch W, Trottenberg U (eds) Multigrid Methods III. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 98. Birkhäuser, Basel – reference: VankaSBlock-implicit multigrid solution of Navier-Stokes equations in primitive variablesJ Comput Phys198665113815884845110.1016/0021-9991(86)90008-20606.76035 – reference: BiotMATheory of elasticity and consolidation for a porous anisotropic solidJ Appl Phys19552621821856687410.1063/1.17219560067.23603 – reference: BrandtAMultigrid solvers for non-elliptic and singular-perturbation steady-state problems1981RehovotThe Weizmann Institute of Science – reference: RodrigoCGasparFHuXZikatanovLStability and monotonicity for some discretizations of the Biot’s consolidation modelComput Methods Appl Mech Eng2016298183204342771110.1016/j.cma.2015.09.019 – reference: WienandsRJoppichWPractical Fourier analysis for multigrid methods2005Boca RatonChapman and Hall/CRC Press1062.65133 – reference: PhillipsPWheelerMOvercoming the problem of locking in linear elasticity and poroelasticity: an heuristic approachComput Geosci200913151210.1007/s10596-008-9114-x1172.74017 – reference: LubichCOstermannAMultigrid dynamic iteration for parabolic equationsBIT19872721623489412410.1007/BF019341860623.65125 – reference: WesselingPAn introduction to multigrid methods1992ChichesterWiley0760.65092 – reference: MacLachlanSPOosterleeCWLocal fourier analysis for multigrid with overlapping smoothers applied to systems of pdesNumer Linear Algebra Appl2011184751774284019810.1002/nla.7621265.65256 – reference: Nordbotten JM (2016) Stable cell-centered finite volume discretization for biot equations. SIAM J Numer Anal 54(2):942–968. https://doi.org/10.1137/15M1014280 – reference: FriedhoffSMacLachlanSA generalized predictive analysis tool for multigrid methodsNumer Linear Algebra Appl2015224618647336782610.1002/nla.19771349.65427 – reference: Aguilar G, Gaspar F, Lisbona F, Rodrigo C (2008) Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int J Numer Meth Eng 75(11):1282–1300. https://doi.org/10.1002/nme.2295 – reference: VandewalleSParallel Multigrid waveform relaxation for parabolic problems1993Stuttgart, B.GTeubner10.1007/978-3-322-94761-10816.65057 – volume-title: Practical Fourier analysis for multigrid methods year: 2005 ident: 603_CR31 – volume-title: Theoretical soil mechanics year: 1943 ident: 603_CR26 doi: 10.1002/9780470172766 – volume: 31 start-page: 1695 issue: 6 year: 1994 ident: 603_CR6 publication-title: SIAM J Numer Anal doi: 10.1137/0731087 – volume-title: Multigrid solvers for non-elliptic and singular-perturbation steady-state problems year: 1981 ident: 603_CR5 – ident: 603_CR15 – volume: 13 start-page: 5 issue: 1 year: 2009 ident: 603_CR23 publication-title: Comput Geosci doi: 10.1007/s10596-008-9114-x – volume-title: Multigrid year: 2001 ident: 603_CR27 – volume: 139 start-page: 274 issue: 2 year: 1998 ident: 603_CR21 publication-title: J Comput Phys doi: 10.1006/jcph.1997.5854 – volume: 65 start-page: 138 issue: 1 year: 1986 ident: 603_CR29 publication-title: J Comput Phys doi: 10.1016/0021-9991(86)90008-2 – volume: 27 start-page: 216 year: 1987 ident: 603_CR17 publication-title: BIT doi: 10.1007/BF01934186 – volume: 11 start-page: 67 issue: 2 year: 2008 ident: 603_CR13 publication-title: Comput Vis Sci doi: 10.1007/s00791-007-0061-1 – ident: 603_CR19 doi: 10.1007/978-3-0348-5712-3_23 – volume: 31 start-page: 333 issue: 138 year: 1977 ident: 603_CR4 publication-title: Math Comput doi: 10.1090/S0025-5718-1977-0431719-X – start-page: 69 volume-title: 50 years of time parallel time integration year: 2015 ident: 603_CR10 – volume: 22 start-page: 618 issue: 4 year: 2015 ident: 603_CR9 publication-title: Numer Linear Algebra Appl doi: 10.1002/nla.1977 – ident: 603_CR14 doi: 10.1002/nag.1062 – volume: 105 start-page: 96 year: 2016 ident: 603_CR25 publication-title: Appl Numer Math doi: 10.1016/j.apnum.2016.02.006 – volume: 229 start-page: 4813 issue: 12 year: 2010 ident: 603_CR8 publication-title: J Comput Phys doi: 10.1016/j.jcp.2010.03.018 – volume: 298 start-page: 183 year: 2016 ident: 603_CR24 publication-title: Comput Methods Appl Mech Eng doi: 10.1016/j.cma.2015.09.019 – ident: 603_CR11 doi: 10.1137/16M1090193 – volume: 26 start-page: 182 issue: 2 year: 1955 ident: 603_CR3 publication-title: J Appl Phys doi: 10.1063/1.1721956 – ident: 603_CR1 doi: 10.1002/nme.2295 – ident: 603_CR12 doi: 10.1016/S0168-9274(02)00190-3 – volume-title: The finite element method in the static and dynamic deformation and consolidation of porous media year: 1998 ident: 603_CR16 – ident: 603_CR20 doi: 10.1137/15M1014280 – volume: 58 start-page: 1933 issue: 12 year: 2008 ident: 603_CR22 publication-title: Appl Numer Math doi: 10.1016/j.apnum.2007.11.014 – volume-title: Parallel Multigrid waveform relaxation for parabolic problems year: 1993 ident: 603_CR28 doi: 10.1007/978-3-322-94761-1 – volume-title: An introduction to multigrid methods year: 1992 ident: 603_CR30 – ident: 603_CR7 doi: 10.1061/9780784412992.110 – volume: 12 start-page: 155 issue: 2 year: 1941 ident: 603_CR2 publication-title: J Appl Phys doi: 10.1063/1.1712886 – volume: 18 start-page: 751 issue: 4 year: 2011 ident: 603_CR18 publication-title: Numer Linear Algebra Appl doi: 10.1002/nla.762
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Snippet	In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This...
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SubjectTerms	Algebra Applications of Mathematics Applied physics Computational mathematics Computational Mathematics and Numerical Analysis Finite difference method Iterative methods Mathematical Applications in Computer Science Mathematical Applications in the Physical Sciences Mathematical models Mathematics Mathematics and Statistics Poroelasticity Relaxation method (mathematics)
Title	A multigrid waveform relaxation method for solving the poroelasticity equations
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