Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations
HTML articles powered by AMS MathViewer
- by Clément Cancès, Flore Nabet and Martin Vohralík HTML | PDF
- Math. Comp. 90 (2021), 517-563 Request permission
Abstract:
We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.References
- Ahmed Ait Hammou Oulhaj, Clément Cancès, and Claire Chainais-Hillairet, Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 4, 1533–1567. MR 3875296, DOI 10.1051/m2an/2017012
- Ahmed Ait Hammou Oulhaj, Clément Cancès, Claire Chainais-Hillairet, and Philippe Laurençot, Large time behavior of a two phase extension of the porous medium equation, Interfaces Free Bound. 21 (2019), no. 2, 199–229. MR 3986535, DOI 10.4171/IFB/421
- Boris Andreianov, Clément Cancès, and Ayman Moussa, A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs, J. Funct. Anal. 273 (2017), no. 12, 3633–3670. MR 3711877, DOI 10.1016/j.jfa.2017.08.010
- Lisa A. Baughman and Noel J. Walkington, Co-volume methods for degenerate parabolic problems, Numer. Math. 64 (1993), no. 1, 45–67. MR 1191322, DOI 10.1007/BF01388680
- J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Vol. 4, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
- Marianne Bessemoulin-Chatard and Claire Chainais-Hillairet, Exponential decay of a finite volume scheme to the thermal equilibrium for drift-diffusion systems, J. Numer. Math. 25 (2017), no. 3, 147–168. MR 3707103, DOI 10.1515/jnma-2016-0007
- M. Bessemoulin-Chatard, C. Chainais-Hillairet, and M.-H. Vignal, Study of a finite volume scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit, SIAM J. Numer. Anal. 52 (2014), no. 4, 1666–1691. MR 3231987, DOI 10.1137/130913432
- Dietrich Braess and Joachim Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp. 77 (2008), no. 262, 651–672. MR 2373174, DOI 10.1090/S0025-5718-07-02080-7
- Konstantin Brenner, Clément Cancès, and Danielle Hilhorst, Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure, Comput. Geosci. 17 (2013), no. 3, 573–597. MR 3050007, DOI 10.1007/s10596-013-9345-3
- K. Brenner and R. Masson, Convergence of a vertex centered discretization of two-phase Darcy flows on general meshes, Int. J. Finite Vol. 10 ([2013]), 37. MR 3276494
- C. Cancès, Energy stable numerical methods for porous media flow type problems, Oil & Gas Science and Technology-Rev. IFPEN 73 (2018), 1–18.
- Clément Cancès and Pascal Omnes (eds.), Finite volumes for complex applications VIII—hyperbolic, elliptic and parabolic problems, Springer Proceedings in Mathematics & Statistics, vol. 200, Springer, Cham, 2017. Papers from the 8th International Symposium (FVCA 8) held in Lille, June 12–16, 2017. MR 3662191, DOI 10.1007/978-3-319-57394-6
- Clément Cancès, Claire Chainais-Hillairet, and Stella Krell, Numerical analysis of a nonlinear free-energy diminishing discrete duality finite volume scheme for convection diffusion equations, Comput. Methods Appl. Math. 18 (2018), no. 3, 407–432. MR 3824772, DOI 10.1515/cmam-2017-0043
- Clément Cancès and Cindy Guichard, Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations, Math. Comp. 85 (2016), no. 298, 549–580. MR 3434871, DOI 10.1090/mcom/2997
- Clément Cancès and Cindy Guichard, Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure, Found. Comput. Math. 17 (2017), no. 6, 1525–1584. MR 3735861, DOI 10.1007/s10208-016-9328-6
- Clément Cancès, Iuliu Sorin Pop, and Martin Vohralík, An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow, Math. Comp. 83 (2014), no. 285, 153–188. MR 3120585, DOI 10.1090/S0025-5718-2013-02723-8
- C. Chainais-Hillairet, Schéma volumes finis pour des problèmes hyperboliques : convergence et estimations d’erreur, Ph.D. Thesis, 1998.
- Claire Chainais-Hillairet and Francis Filbet, Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model, IMA J. Numer. Anal. 27 (2007), no. 4, 689–716. MR 2371828, DOI 10.1093/imanum/drl045
- Marianne Chatard, Asymptotic behavior of the Scharfetter-Gummel scheme for the drift-diffusion model, Finite volumes for complex applications VI. Problems & perspectives. Volume 1, 2, Springer Proc. Math., vol. 4, Springer, Heidelberg, 2011, pp. 235–243. MR 2815643, DOI 10.1007/978-3-642-20671-9_{2}5
- G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, Stud. Math. Appl., Vol. 17, North-Holland, Amsterdam, 1986.
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Philippe Destuynder and Brigitte Métivet, Explicit error bounds in a conforming finite element method, Math. Comp. 68 (1999), no. 228, 1379–1396. MR 1648383, DOI 10.1090/S0025-5718-99-01093-5
- Daniele A. Di Pietro, Martin Vohralík, and Soleiman Yousef, Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem, Math. Comp. 84 (2015), no. 291, 153–186. MR 3266956, DOI 10.1090/S0025-5718-2014-02854-8
- Vít Dolejší, Alexandre Ern, and Martin Vohralík, A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems, SIAM J. Numer. Anal. 51 (2013), no. 2, 773–793. MR 3033032, DOI 10.1137/110859282
- A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Series, vol. 159, Springer, New York, 2004.
- Alexandre Ern, Iain Smears, and Martin Vohralík, Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems, SIAM J. Numer. Anal. 55 (2017), no. 6, 2811–2834. MR 3723331, DOI 10.1137/16M1097626
- Alexandre Ern and Martin Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput. 35 (2013), no. 4, A1761–A1791. MR 3072765, DOI 10.1137/120896918
- Alexandre Ern and Martin Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal. 53 (2015), no. 2, 1058–1081. MR 3335498, DOI 10.1137/130950100
- Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
- Robert Eymard, Cindy Guichard, Raphaele Herbin, and Roland Masson, Vertex-centred discretization of multiphase compositional Darcy flows on general meshes, Comput. Geosci. 16 (2012), no. 4, 987–1005. MR 2978830, DOI 10.1007/s10596-012-9299-x
- Robert Eymard, Cindy Guichard, Raphaèle Herbin, and Roland Masson, Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 7-8, 560–585. MR 3230270, DOI 10.1002/zamm.201200206
- Robert Eymard, Michaël Gutnic, and Danielle Hilhorst, The finite volume method for Richards equation, Comput. Geosci. 3 (1999), no. 3-4, 259–294 (2000). MR 1750075, DOI 10.1023/A:1011547513583
- Robert Eymard, Raphaèle Herbin, and Anthony Michel, Mathematical study of a petroleum-engineering scheme, M2AN Math. Model. Numer. Anal. 37 (2003), no. 6, 937–972. MR 2026403, DOI 10.1051/m2an:2003062
- Robert Eymard, Danielle Hilhorst, and Martin Vohralík, A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numer. Math. 105 (2006), no. 1, 73–131. MR 2257386, DOI 10.1007/s00211-006-0036-z
- André Fiebach, Annegret Glitzky, and Alexander Linke, Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems, Numer. Math. 128 (2014), no. 1, 31–72. MR 3248048, DOI 10.1007/s00211-014-0604-6
- André Fiebach, Annegret Glitzky, and Alexander Linke, Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems, Numer. Methods Partial Differential Equations 32 (2016), no. 1, 141–174. MR 3434622, DOI 10.1002/num.21990
- Francis Filbet and Maxime Herda, A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure, Numer. Math. 137 (2017), no. 3, 535–577. MR 3712285, DOI 10.1007/s00211-017-0885-7
- Peter A. Forsyth, A control volume finite element approach to NAPL groundwater contamination, SIAM J. Sci. Statist. Comput. 12 (1991), no. 5, 1029–1057. MR 1114973, DOI 10.1137/0912055
- F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
- Raphaèle Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Numer. Methods Partial Differential Equations 11 (1995), no. 2, 165–173. MR 1316144, DOI 10.1002/num.1690110205
- Joseph W. Jerome and Michael E. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), no. 160, 377–414. MR 669635, DOI 10.1090/S0025-5718-1982-0669635-2
- Anvarbek M. Meirmanov, The Stefan problem, De Gruyter Expositions in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1992. Translated from the Russian by Marek Niezgódka and Anna Crowley; With an appendix by the author and I. G. Götz. MR 1154310, DOI 10.1515/9783110846720.245
- Anthony Michel, A finite volume scheme for two-phase immiscible flow in porous media, SIAM J. Numer. Anal. 41 (2003), no. 4, 1301–1317. MR 2034882, DOI 10.1137/S0036142900382739
- A. Moussa, Some variants of the classical Aubin-Lions lemma, J. Evol. Equ. 16 (2016), no. 1, 65–93. MR 3466213, DOI 10.1007/s00028-015-0293-3
- R. H. Nochetto, M. Paolini, and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates, Math. Comp. 57 (1991), no. 195, 73–108, S1–S11. MR 1079028, DOI 10.1090/S0025-5718-1991-1079028-X
- R. H. Nochetto, A. Schmidt, and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp. 69 (2000), no. 229, 1–24. MR 1648399, DOI 10.1090/S0025-5718-99-01097-2
- Felix Otto, $L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996), no. 1, 20–38. MR 1415045, DOI 10.1006/jdeq.1996.0155
- Benoît Perthame, Parabolic equations in biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2015. Growth, reaction, movement and diffusion. MR 3408563, DOI 10.1007/978-3-319-19500-1
- Jim Rulla and Noel J. Walkington, Optimal rates of convergence for degenerate parabolic problems in two dimensions, SIAM J. Numer. Anal. 33 (1996), no. 1, 56–67. MR 1377243, DOI 10.1137/0733004
- Juan Luis Vázquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. MR 2286292
- Augusto Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and their Applications, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1423808, DOI 10.1007/978-1-4612-4078-5
Additional Information
- Clément Cancès
- Affiliation: Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France
- Email: clement.cances@inria.fr
- Flore Nabet
- Affiliation: CMAP, Ecole polytechnique, CNRS, I.P. Paris, 91128 Palaiseau, France
- MR Author ID: 1083928
- ORCID: 0000-0001-7828-251X
- Email: flore.nabet@polytechnique.edu
- Martin Vohralík
- Affiliation: Inria, 2 rue Simone Iff, 75589 Paris, France; and Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France
- ORCID: 0000-0002-8838-7689
- Email: martin.vohralik@inria.fr
- Received by editor(s): October 16, 2018
- Received by editor(s) in revised form: February 13, 2020
- Published electronically: November 5, 2020
- Additional Notes: This work was supported by the French National Research Agency ANR (project GeoPor, grant ANR-13-JS01-0007-01). The third author has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR)
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 517-563
- MSC (2010): Primary 65M12, 35K65, 65M15, 65M60
- DOI: https://doi.org/10.1090/mcom/3577
- MathSciNet review: 4194153