Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem
HTML articles powered by AMS MathViewer
- by Daniele A. Di Pietro, Martin Vohralík and Soleiman Yousef PDF
- Math. Comp. 84 (2015), 153-186 Request permission
Abstract:
We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the $L^2(L^2)$ error of the temperature and the $L^2(H^{-1})$ error of the enthalpy. Moreover, they allow us to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.References
- Hans Wilhelm Alt and Stephan Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), no. 3, 311–341. MR 706391, DOI 10.1007/BF01176474
- Geneviève Amiez and Pierre-Alain Gremaud, On a numerical approach to Stefan-like problems, Numer. Math. 59 (1991), no. 1, 71–89. MR 1103754, DOI 10.1007/BF01385771
- B. Andreianov, M. Bendahmane, and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperbolic Differ. Equ. 7 (2010), no. 1, 1–67. MR 2646796, DOI 10.1142/S0219891610002062
- 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
- M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751–756. MR 2036927, DOI 10.4171/ZAA/1170
- G. Beckett, J. A. Mackenzie, and M. L. Robertson, A moving mesh finite element method for the solution of two-dimensional Stefan problems, J. Comput. Phys. 168 (2001), no. 2, 500–518. MR 1826524, DOI 10.1006/jcph.2001.6721
- Philippe Benilan and Petra Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations 1 (1996), no. 6, 1053–1073. MR 1409899
- 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
- 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
- Libor Čermák and Miloš Zlámal, Transformation of dependent variables and the finite element solution of nonlinear evolution equations, Internat. J. Numer. Methods Engrg. 15 (1980), no. 1, 31–40. MR 554438, DOI 10.1002/nme.1620150104
- 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
- J. F. Ciavaldini, Analyse numerique d’un problème de Stefan à deux phases par une methode d’éléments finis, SIAM J. Numer. Anal. 12 (1975), 464–487 (French, with English summary). MR 391741, DOI 10.1137/0712037
- P. Destuynder and B. Métivet. Explicit error bounds in a conforming finite element method. Math. Comp., 68(228):1379–1396, 1999.
- 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
- Linda El Alaoui, Alexandre Ern, and Martin Vohralík, Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 37-40, 2782–2795. MR 2811915, DOI 10.1016/j.cma.2010.03.024
- Charles M. Elliott, On the finite element approximation of an elliptic variational inequality arising from an implicit time discretization of the Stefan problem, IMA J. Numer. Anal. 1 (1981), no. 1, 115–125. MR 607251, DOI 10.1093/imanum/1.1.115
- Alexandre Ern and Martin Vohralík, Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids, C. R. Math. Acad. Sci. Paris 347 (2009), no. 7-8, 441–444 (English, with English and French summaries). MR 2537245, DOI 10.1016/j.crma.2009.01.017
- Alexandre Ern and Martin Vohralík, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal. 48 (2010), no. 1, 198–223. MR 2608366, DOI 10.1137/090759008
- 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
- 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
- R. Eymard, T. Gallouët, D. Hilhorst, and Y. Naït Slimane, Finite volumes and nonlinear diffusion equations, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 6, 747–761 (English, with English and French summaries). MR 1652593, DOI 10.1051/m2an/1998320607471
- Avner Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51–87. MR 227625, DOI 10.1090/S0002-9947-1968-0227625-7
- Danielle Hilhorst and Martin Vohralík, A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 5-8, 597–613. MR 2749021, DOI 10.1016/j.cma.2010.08.017
- Willi Jäger and Jozef Kačur, Solution of porous medium type systems by linear approximation schemes, Numer. Math. 60 (1991), no. 3, 407–427. MR 1137200, DOI 10.1007/BF01385729
- 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
- Pavel Jiránek, Zdeněk Strakoš, and Martin Vohralík, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput. 32 (2010), no. 3, 1567–1590. MR 2652091, DOI 10.1137/08073706X
- S. L. Kamenomostskaja, On Stefan’s problem, Mat. Sb. (N.S.) 53 (95) (1961), 489–514 (Russian). MR 0141895
- C. T. Kelley and Jim Rulla, Solution of the time discretized Stefan problem by Newton’s method, Nonlinear Anal. 14 (1990), no. 10, 851–872. MR 1055534, DOI 10.1016/0362-546X(90)90025-C
- P. Ladevèze. Comparaison de modèles de milieux continus. Ph.D. thesis, Université Pierre et Marie Curie (Paris 6), 1975.
- R. Luce and B. I. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal. 42 (2004), no. 4, 1394–1414. MR 2114283, DOI 10.1137/S0036142903433790
- Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, DOI 10.1137/S0036142902406314
- Gunter H. Meyer, Multidimensional Stefan problems, SIAM J. Numer. Anal. 10 (1973), 522–538. MR 331807, DOI 10.1137/0710047
- Ricardo H. Nochetto, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math. 4 (1987), no. 1, 111–138. MR 899207, DOI 10.1007/BF03167758
- 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, M. Paolini, and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. II. Implementation and numerical experiments, SIAM J. Sci. Statist. Comput. 12 (1991), no. 5, 1207–1244. MR 1114983, DOI 10.1137/0912065
- 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
- R. H. Nochetto and C. Verdi, The combined use of a nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numer. Funct. Anal. Optim. 9 (1987/88), no. 11-12, 1177–1192. MR 936337, DOI 10.1080/01630568808816279
- 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
- L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
- M. Picasso, An adaptive finite element algorithm for a two-dimensional stationary Stefan-like problem, Comput. Methods Appl. Mech. Engrg. 124 (1995), no. 3, 213–230. MR 1343078, DOI 10.1016/0045-7825(95)00793-Z
- Marco Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3-4, 223–237. MR 1673951, DOI 10.1016/S0045-7825(98)00121-2
- I. S. Pop, M. Sepúlveda, F. A. Radu, and O. P. Vera Villagrán. Error estimates for the finite volume discretization for the porous medium equation. J. Comput. Appl. Math., 234(7):2135–2142, 2010.
- W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
- Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729, DOI 10.1007/978-3-540-85268-1
- Sergey Repin, A posteriori estimates for partial differential equations, Radon Series on Computational and Applied Mathematics, vol. 4, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2458008, DOI 10.1515/9783110203042
- R. Verfürth, A posteriori error estimates for nonlinear problems. $L^r(0,T;L^\rho (\Omega ))$-error estimates for finite element discretizations of parabolic equations, Math. Comp. 67 (1998), no. 224, 1335–1360. MR 1604371, DOI 10.1090/S0025-5718-98-01011-4
- R. Verfürth, A posteriori error estimates for nonlinear problems: $L^r(0,T;W^{1,\rho }(\Omega ))$-error estimates for finite element discretizations of parabolic equations, Numer. Methods Partial Differential Equations 14 (1998), no. 4, 487–518. MR 1627578, DOI 10.1002/(SICI)1098-2426(199807)14:4<487::AID-NUM4>3.0.CO;2-G
- R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, Calcolo 40 (2003), no. 3, 195–212. MR 2025602, DOI 10.1007/s10092-003-0073-2
- R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1766–1782. MR 2182149, DOI 10.1137/040604261
- Martin Vohralík, A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization, C. R. Math. Acad. Sci. Paris 346 (2008), no. 11-12, 687–690 (English, with English and French summaries). MR 2423279, DOI 10.1016/j.crma.2008.03.006
- Martin Vohralík, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, J. Sci. Comput. 46 (2011), no. 3, 397–438. MR 2765501, DOI 10.1007/s10915-010-9410-1
- J. A. Wheeler, Permafrost design for the trans-Alaska, in P. T. Boggs, editor, Moving Boundary Problems, Academic Press, New York, 1978.
- Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI 10.1137/0710062
Additional Information
- Daniele A. Di Pietro
- Affiliation: Université Montpellier 2, I3M, 34057 Montpellier, France
- Email: daniele.di-pietro@univ-montp2.fr
- Martin Vohralík
- Affiliation: UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France – and – CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France
- Address at time of publication: INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France
- ORCID: 0000-0002-8838-7689
- Email: martin.vohralik@inria.fr
- Soleiman Yousef
- Affiliation: UPMC Université, Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France – and – IFP Energies nouvelles, Department of Applied Mathematics, 1 & 4 avenue Bois Préau, 92852 Rueil-Malmaison, France
- Email: yousef@ann.jussieu.fr
- Received by editor(s): April 24, 2012
- Received by editor(s) in revised form: April 27, 2013
- Published electronically: June 18, 2014
- Additional Notes: This work was supported by the ERT project “Enhanced oil recovery and geological sequestration of $\mathrm {CO}_2$: mesh adaptivity, a posteriori error control, and other advanced techniques” (LJLL/IFPEN)
The third author is the corresponding author - © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 153-186
- MSC (2010): Primary 65N08, 65N15, 65N50, 80A22
- DOI: https://doi.org/10.1090/S0025-5718-2014-02854-8
- MathSciNet review: 3266956