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An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

Authors: Clément Cancès, Iuliu Sorin Pop and Martin Vohralík
Journal: Math. Comp. 83 (2014), 153-188
MSC (2010): Primary 65M15, 76S05, 76T99, 65M08
Published electronically: June 28, 2013
MathSciNet review: 3120585
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Abstract: In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ``mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ``engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient.

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  • 1. M. AINSWORTH AND J. T. ODEN, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
  • 2. G. AKRIVIS, C. MAKRIDAKIS, AND R. H. NOCHETTO, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp., 75 (2006), pp. 511-531. MR 2196979 (2007a:65114)
  • 3. H. W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), pp. 311-341. MR 706391 (85c:35059)
  • 4. L. ANGERMANN, P. KNABNER, AND K. THIELE, An error estimator for a finite volume discretization of density driven flow in porous media, Appl. Numer. Math., 26 (1998), pp. 179-191.
    Proceedings of the International Centre for Mathematical Sciences Conference on Grid Adaptation in Computational PDEs: Theory and Applications (Edinburgh, 1996). MR 1602860 (98m:76116)
  • 5. S. N. ANTONTSEV, A. V. KAZHIKHOV, AND V. N. MONAKHOV, Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland, Amsterdam, 1990.
    Studies in Mathematics and Its Applications, Vol. 22. MR 1035212 (91d:76018)
  • 6. T. ARBOGAST, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), pp. 1009-1031. MR 1194142 (93k:76107)
  • 7. J. BEAR, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
  • 8. J. BEAR AND Y. BACHMAT, Introduction to Modeling of Transport Phenomena in Porous Media, vol. 4 of Theory and Applications of Transport in Porous Media, Kluwer Academic Publishers, Dordrecht, Holland, 1990.
  • 9. F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 10. C. CANCèS AND T. GALLOUëT, On the time continuity of entropy solutions, J. Evol. Equ., 11 (2011). MR 2780572 (2012b:35140)
  • 11. J. CARRILLO, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), pp. 269-361. MR 1709116 (2000m:35132)
  • 12. G. CHAVENT AND J. JAFFRé, Mathematical models and finite elements for reservoir simulation, North-Holland, Amsterdam, 1986.
    Studies in Mathematics and Its Applications, Vol. 17.
  • 13. Y. CHEN AND W. LIU, A posteriori error estimates of mixed methods for miscible displacement problems, Internat. J. Numer. Methods Engrg., 73 (2008), pp. 331-343. MR 2382047 (2009b:76157)
  • 14. Z. CHEN, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution, J. Differential Equations, 171 (2001), pp. 203-232. MR 1818648 (2002a:76158)
  • 15. -, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization, J. Differential Equations, 186 (2002), pp. 345-376. MR 1942213
  • 16. Z. CHEN AND R. E. EWING, Degenerate two-phase incompressible flow. III. Sharp error estimates, Numer. Math., 90 (2001), pp. 215-240. MR 1872726 (2002j:76128)
  • 17. -, Degenerate two-phase incompressible flow. IV. Local refinement and domain decomposition, J. Sci. Comput., 18 (2003), pp. 329-360. MR 1967254
  • 18. Z. CHEN AND G. JI, Sharp $ L\sp 1$ a posteriori error analysis for nonlinear convection-diffusion problems, Math. Comp., 75 (2006), pp. 43-71. MR 2176389 (2006h:65134)
  • 19. J. DE FRUTOS, B. GARCíA-ARCHILLA, AND J. NOVO, A posteriori error estimates for fully discrete nonlinear parabolic problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 3462-3474. MR 2335276 (2008f:65161)
  • 20. D. A. DPIETRO, M. VOHRALíK, AND C. WIDMER, An a posteriori error estimator for a finite volume discretization of the two-phase flow, in Finite Volumes for Complex Applications VI, J. Fořt, J. Fürst, J. Halama, R. Herbin, and F. Hubert, eds., Berlin, Heidelberg, 2011, Springer-Verlag, pp. 341-349. MR 2882311
  • 21. D. A. DPIETRO, M. VOHRALíK, AND S. YOUSEF, Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem. HAL Preprint 00690862, submitted for publication, 2012.
  • 22. V. DOLEJŠí, A. ERN, AND M. 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. DOI 10.1137/110859282. MR 3033032
  • 23. L. EALAOUI, A. ERN, AND M. 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), pp. 597-613. MR 2811915 (2012e:65267)
  • 24. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), pp. 1729-1749. MR 1360457 (96i:65081)
  • 25. A. ERN AND M. VOHRAL´İK, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal., 48 (2010), pp. 198-223. MR 2608366 (2011d:65281)
  • 26. A. ERN AND M. VOHRAL´İK, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., Accepted for publication, 2013.
  • 27. R. EYMARD AND T. GALLOUëT, Convergence d'un schéma de type éléments finis-volumes finis pour un système formé d'une équation elliptique et d'une équation hyperbolique, RAIRO Modél. Math. Anal. Numér., 27 (1993), pp. 843-861. MR 1249455 (95b:65118)
  • 28. R. EYMARD, T. GALLOUëT, AND R. HERBIN, Finite volume methods, in Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam, 2000, pp. 713-1020. MR 1804748 (2002e:65138)
  • 29. R. EYMARD, R. HERBIN, AND A. MICHEL, Mathematical study of a petroleum-engineering scheme, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 937-972. MR 2026403 (2004j:76147)
  • 30. L. GALLIMARD, P. LADEVèZE, AND J. P. PELLE, Error estimation and time-space parameters optimization for FEM non-linear computation, Computers & Structures, 64 (1997), pp. 145-156.
  • 31. R. HUBER AND R. HELMIG, Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media, Comput. Geosci., 4 (2000), pp. 141-164. MR 1800561 (2001j:76073)
  • 32. W. JäGER AND J. KAČUR, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, RAIRO Modél. Math. Anal. Numér., 29 (1995), pp. 605-627. MR 1352864 (96g:65087)
  • 33. P. JIRáNEK, Z. STRAKOŠ, AND M. VOHRALíK, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput., 32 (2010), pp. 1567-1590. MR 2652091 (2011e:65233)
  • 34. D. KRöNER AND S. LUCKHAUS, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), pp. 276-288. MR 764127 (85k:35211)
  • 35. S. N. KRUŽKOV, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (123) (1970), pp. 228-255. MR 0267257 (42:2159)
  • 36. A. MICHEL, A finite volume scheme for two-phase immiscible flow in porous media, SIAM J. Numer. Anal., 41 (2003), pp. 1301-1317. MR 2034882 (2005a:65087)
  • 37. P. NEITTAANMäKI AND S. REPIN, Reliable methods for computer simulation, vol. 33 of Studies in Mathematics and its Applications, Elsevier Science B.V., Amsterdam, 2004.
    Error control and a posteriori estimates. MR 2095603 (2005k:65005)
  • 38. R. H. NOCHETTO, G. SAVARé, AND C. VERDI, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., 53 (2000), pp. 525-589. MR 1737503 (2000k:65142)
  • 39. R. H. NOCHETTO AND C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), pp. 784-814. MR 954786 (89m:65102)
  • 40. M. OHLBERGER, A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations, Numer. Math., 87 (2001), pp. 737-761. MR 1815733 (2001m:65120)
  • 41. -, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 355-387. MR 1825703 (2002a:65142)
  • 42. F. OTTO, $ L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations, 131 (1996), pp. 20-38. MR 1415045 (97i:35125)
  • 43. I. S. POP, Error estimates for a time discretization method for the Richards' equation, Comput. Geosci., 6 (2002), pp. 141-160. MR 1926564 (2003h:65136)
  • 44. I. S. POP, F. RADU, AND P. KNABNER, Mixed finite elements for the Richards' equation: linearization procedure, J. Comput. Appl. Math., 168 (2004), pp. 365-373. MR 2079503
  • 45. W. PRAGER AND J. L. SYNGE, Approximations in elasticity based on the concept of function space, Quart. Appl. Math., 5 (1947), pp. 241-269. MR 0025902 (10:81b)
  • 46. A. QUARTERONI AND A. VALLI, Numerical approximation of partial differential equations, vol. 23 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1994. MR 1299729 (95i:65005)
  • 47. F. A. RADU, I. S. POP, AND P. KNABNER, Newton-type methods for the mixed finite element discretization of some degenerate parabolic equations, in Numerical mathematics and advanced applications, Springer, Berlin, 2006, pp. 1192-1200. MR 2303752
  • 48. height 2pt depth -1.6pt width 23pt, Error estimates for a mixed finite element discretization of some degenerate parabolic equations, Numer. Math., 109 (2008), pp. 285-311. MR 2385655
  • 49. S. I. REPIN, A posteriori estimates for partial differential equations, vol. 4 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2458008 (2010b:35004)
  • 50. J. E. ROBERTS AND J.-M. THOMAS, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 523-639. MR 1115239
  • 51. R. VERFüRTH, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Teubner-Wiley, Stuttgart, 1996.
  • 52. -, 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), pp. 487-518. MR 1627578 (99g:65099)
  • 53. -, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal., 43 (2005), pp. 1766-1782. MR 2182149 (2007d:65116)
  • 54. M. VOHRALíK, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, J. Sci. Comput., 46 (2011), pp. 397-438. MR 2765501 (2012a:65297)
  • 55. M. VOHRALíK AND M. F. WHEELER, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, HAL Preprint 00633594v2, submitted for publication, 2013.

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Additional Information

Clément Cancès
Affiliation: LJLL – UPMC Paris 06, Boite Courrier 187, 4 place Jussieu, 75005 Paris, France

Iuliu Sorin Pop
Affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB, Eindhoven, the Netherlands

Martin Vohralík
Affiliation: LJLL – UPMC Paris 06, Boite Courrier 187, 4 place Jussieu, 75005 Paris, France
Address at time of publication: INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France

Received by editor(s): September 13, 2011
Received by editor(s) in revised form: April 25, 2012
Published electronically: June 28, 2013
Additional Notes: This work was partly supported by the Groupement MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN) and 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)
Article copyright: © Copyright 2013 American Mathematical Society

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