A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s model
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- by Arbaz Khan and Pietro Zanotti HTML | PDF
- Math. Comp. 91 (2022), 1143-1170 Request permission
Abstract:
We consider the system of partial differential equations stemming from the time discretization of the two-field formulation of the Biot’s model with the backward Euler scheme. A typical difficulty encountered in the space discretization of this problem is the robustness with respect to various material parameters. We deal with this issue by observing that the problem is uniformly stable, irrespective of all parameters, in a suitable nonsymmetric variational setting. Guided by this result, we design a novel nonconforming discretization, which employs Crouzeix-Raviart and discontinuous elements. We prove that the proposed discretization is quasi-optimal and robust in a parameter-dependent norm and discuss the consequences of this result.References
- Douglas N. Arnold, On nonconforming linear-constant elements for some variants of the Stokes equations, Istit. Lombardo Accad. Sci. Lett. Rend. A 127 (1993), no. 1, 83–93 (1994) (English, with Italian summary). MR 1284845
- Sören Bartels and Zhangxian Wang, Orthogonality relations of Crouzeix-Raviart and Raviart-Thomas finite element spaces, Numer. Math. 148 (2021), no. 1, 127–139. MR 4265901, DOI 10.1007/s00211-021-01199-3
- Lorenz Berger, Rafel Bordas, David Kay, and Simon Tavener, Stabilized lowest-order finite element approximation for linear three-field poroelasticity, SIAM J. Sci. Comput. 37 (2015), no. 5, A2222–A2245. MR 3394360, DOI 10.1137/15M1009822
- Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
- Susanne C. Brenner, Korn’s inequalities for piecewise $H^1$ vector fields, Math. Comp. 73 (2004), no. 247, 1067–1087. MR 2047078, DOI 10.1090/S0025-5718-03-01579-5
- C. Carstensen and M. Schedensack, Medius analysis and comparison results for first-order finite element methods in linear elasticity, IMA J. Numer. Anal. 35 (2015), no. 4, 1591–1621. MR 3407237, DOI 10.1093/imanum/dru048
- Yumei Chen, Yan Luo, and Minfu Feng, Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem, Appl. Math. Comput. 219 (2013), no. 17, 9043–9056. MR 3047799, DOI 10.1016/j.amc.2013.03.104
- Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
- Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
- Thirupathi Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169–2189. MR 2684360, DOI 10.1090/S0025-5718-10-02360-4
- J. B. Haga, H. Osnes, and H. P. Langtangen, On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int. J. Numer. Anal. Methods Geomech. 36 (2012), 1507–1522.
- Peter Hansbo and Mats G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, M2AN Math. Model. Numer. Anal. 37 (2003), no. 1, 63–72. MR 1972650, DOI 10.1051/m2an:2003020
- Qingguo Hong and Johannes Kraus, Parameter-robust stability of classical three-field formulation of Biot’s consolidation model, Electron. Trans. Numer. Anal. 48 (2018), 202–226. MR 3820123, DOI 10.1553/etna_{v}ol48s202
- Xiaozhe Hu, Carmen Rodrigo, Francisco J. Gaspar, and Ludmil T. Zikatanov, A nonconforming finite element method for the Biot’s consolidation model in poroelasticity, J. Comput. Appl. Math. 310 (2017), 143–154. MR 3544596, DOI 10.1016/j.cam.2016.06.003
- A. Khan and P. Zanotti, A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s consolidation model. Part II – numerical aspects, In preparation.
- Johannes Korsawe and Gerhard Starke, A least-squares mixed finite element method for Biot’s consolidation problem in porous media, SIAM J. Numer. Anal. 43 (2005), no. 1, 318–339. MR 2177147, DOI 10.1137/S0036142903432929
- Sarvesh Kumar, Ricardo Oyarzúa, Ricardo Ruiz-Baier, and Ruchi Sandilya, Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity, ESAIM Math. Model. Numer. Anal. 54 (2020), no. 1, 273–299. MR 4058208, DOI 10.1051/m2an/2019063
- Jeonghun J. Lee, Robust error analysis of coupled mixed methods for Biot’s consolidation model, J. Sci. Comput. 69 (2016), no. 2, 610–632. MR 3551338, DOI 10.1007/s10915-016-0210-0
- Jeonghun J. Lee, Kent-Andre Mardal, and Ragnar Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput. 39 (2017), no. 1, A1–A24. MR 3590654, DOI 10.1137/15M1029473
- Kent-Andre Mardal, Marie E. Rognes, and Travis B. Thompson, Accurate discretization of poroelasticity without Darcy stability: Stokes-Blot stability revisited, BIT 61 (2021), no. 3, 941–976. MR 4292453, DOI 10.1007/s10543-021-00849-0
- Jan Martin Nordbotten, Stable cell-centered finite volume discretization for Biot equations, SIAM J. Numer. Anal. 54 (2016), no. 2, 942–968. MR 3478962, DOI 10.1137/15M1014280
- Ricardo Oyarzúa and Ricardo Ruiz-Baier, Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal. 54 (2016), no. 5, 2951–2973. MR 3552204, DOI 10.1137/15M1050082
- Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case, Comput. Geosci. 11 (2007), no. 2, 145–158. MR 2327966, DOI 10.1007/s10596-007-9044-z
- Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci. 12 (2008), no. 4, 417–435. MR 2461315, DOI 10.1007/s10596-008-9082-1
- C. Rodrigo, F. J. Gaspar, X. Hu, and L. T. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model, Comput. Methods Appl. Mech. Engrg. 298 (2016), 183–204. MR 3427711, DOI 10.1016/j.cma.2015.09.019
- C. Rodrigo, X. Hu, P. Ohm, J. H. Adler, F. J. Gaspar, and L. T. Zikatanov, New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Engrg. 341 (2018), 467–484. MR 3845633, DOI 10.1016/j.cma.2018.07.003
- Andreas Veeser and Pietro Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. I—Abstract theory, SIAM J. Numer. Anal. 56 (2018), no. 3, 1621–1642. MR 3816182, DOI 10.1137/17M1116362
- Andreas Veeser and Pietro Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. II—Overconsistency and classical nonconforming elements, SIAM J. Numer. Anal. 57 (2019), no. 1, 266–292. MR 3907927, DOI 10.1137/17M1151651
- Andreas Veeser and Pietro Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. III—Discontinuous Galerkin and other interior penalty methods, SIAM J. Numer. Anal. 56 (2018), no. 5, 2871–2894. MR 3857891, DOI 10.1137/17M1151675
- Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. MR 3059294, DOI 10.1093/acprof:oso/9780199679423.001.0001
- Son-Young Yi, A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model, Numer. Methods Partial Differential Equations 29 (2013), no. 5, 1749–1777. MR 3092329, DOI 10.1002/num.21775
- Son-Young Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model, Numer. Methods Partial Differential Equations 30 (2014), no. 4, 1189–1210. MR 3200272, DOI 10.1002/num.21865
Additional Information
- Arbaz Khan
- Affiliation: Department of Mathematics, Indian Institute of Technology Roorkee (IITR), 247667 Roorkee, India
- MR Author ID: 1108913
- ORCID: 0000-0001-6625-700X
- Email: arbaz@ma.iitr.ac.in
- Pietro Zanotti
- Affiliation: Dipartimento di Matematica ‘F. Enriques’, Università degli Studi di Milano, 20133 Milano, Italy
- Address at time of publication: Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia, 27100 Pavia, Italy
- MR Author ID: 1275475
- ORCID: 0000-0003-4505-3520
- Email: pietro.zanotti@unipv.it
- Received by editor(s): August 12, 2020
- Received by editor(s) in revised form: May 23, 2021, and September 2, 2021
- Published electronically: December 1, 2021
- Additional Notes: The first author was supported by the SERB MATRICS grant MTR/2020/000303. The second author was supported by the INdAM-GNCS through the program “Finanziamento giovani ricercatori 2019-2020” and by the MIUR-PRIN 2017 project “Numerical analysis of full and reduced order methods for partial differential equations”
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 1143-1170
- MSC (2020): Primary 65N30, 65N12, 65N15, 76S05
- DOI: https://doi.org/10.1090/mcom/3699
- MathSciNet review: 4405491