×

A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation. (English) Zbl 1259.65132

Summary: We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection-reaction-diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet-Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
35K65 Degenerate parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76M12 Finite volume methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, R.A., Fournier, J.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140. Academic Press, New York (2003)
[2] Afif, M., Amaziane, B.: On convergence of finite volume schemes for a degenerate convection–diffusion equation arising in flow in porous media. Comput. Methods Appl. Mech. Eng. 191, 5265–5286 (2002) · Zbl 1012.76057 · doi:10.1016/S0045-7825(02)00458-9
[3] Angelini, O., Chavant, C., Chénier, E., Eymard, R.: A finite volume scheme for diffusion problems on general meshes applying monotony constraints. SIAM J. Numer. Anal. 47, 4193–4213 (2010) · Zbl 1261.65108 · doi:10.1137/080732183
[4] Arbogast, T., Wheeler, M.F., Zhang, N.: A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33, 1669–1687 (1996) · Zbl 0856.76033 · doi:10.1137/S0036142994266728
[5] Baughman, L., Walkington, N.: Co-volume methods for degenerate parabolic problems. Numer. Math. 64, 45–67 (1993) · Zbl 0797.65075 · doi:10.1007/BF01388680
[6] Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. Reidel, Dordrecht (1987)
[7] Beirão Da Veiga, L., Droniou, J., Manzini, G.: A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems. IMA J. Numer. Anal. 1–45 (2010). doi: 10.1093/imanum/drq018 · Zbl 1263.65102
[8] Beirão Da Veiga, L., Lipnikov, K., Manzini, G.: Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113(3), 325–356 (2009) · Zbl 1183.65132 · doi:10.1007/s00211-009-0234-6
[9] Berndt, M., Lipnikov, K., Moulton, J.D., Shashkov, M.: Convergence of mimetic finite difference discretizations of the diffusion equation. East-West J Numer. Math. 9, 253–284 (2001) · Zbl 1014.65114
[10] Brenner, K.: Hybrid finite volume scheme for a two-phase flow problem (2012, to appear) · Zbl 1357.76045
[11] Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. Math. Model Numer. Anal. 43, 277–295 (2009) · Zbl 1177.65164 · doi:10.1051/m2an:2008046
[12] Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005) · Zbl 1108.65102 · doi:10.1137/040613950
[13] Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005) · Zbl 1083.65099 · doi:10.1142/S0218202505000832
[14] Chainais-Hillairet, C., Droniou, J.: Convergence analysis of a mixed finite volume scheme for an elliptic–parabolic system modeling miscible fluid flows in porous media. SIAM J. Numer. Anal. 45(5), 2228–2258 (2007) · Zbl 1146.76034 · doi:10.1137/060657236
[15] Dawson, C.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35, 1709–1724 (1998) · Zbl 0954.76043 · doi:10.1137/S0036142993259421
[16] Dawson, C., Aizinger, V.: Upwind mixed methods for transport equations. Comput. Geosci. 3, 93–110 (1999) · Zbl 0962.65084 · doi:10.1023/A:1011531109949
[17] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) · Zbl 0559.47040
[18] Droniou, J.: Remarks on discretizations of convection terms in hybrid mimetic mixed methods. Netw. Heterog. Media 5, 3 (2010) · Zbl 1260.76017 · doi:10.3934/nhm.2010.5.545
[19] Droniou, J., Eymard, R.: Study of the mixed finite volume method for stokes and Navier–Stokes equations. Numer. Meth. P. D. E. 25(1), 137–171 (2009) · Zbl 1153.76044 · doi:10.1002/num.20333
[20] Droniou, J., Eymard R., Gallouët, T., Herbin, R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. M3AS 20(2), 265–295 (2010) · Zbl 1191.65142
[21] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 7. Elsevier Science B.V., Amsterdam (2000) · Zbl 0981.65095
[22] Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010) · Zbl 1202.65144 · doi:10.1093/imanum/drn084
[23] Eymard, R., Gallouët, T., Herbin, R., Michel, A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 (2002) · Zbl 1005.65099 · doi:10.1007/s002110100342
[24] Eymard, R., Hilhorst, D., Vohralík, M.: A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105, 73–131 (2006) · Zbl 1108.65099 · doi:10.1007/s00211-006-0036-z
[25] Eymard, R., Hilhorst, D., Vohralík, M.: A combined finite volume-finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids. Numer. Methods Partial Differ. Equ. 26, 612–646 (2009) · Zbl 1192.65117
[26] Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids for anisotropic heterogeneous diffusion problems. In: Finite Volumes for Complex Applications V, pp. 659–692 (2008) · Zbl 1422.65314
[27] Kačur, J.: Solution of degenerate convection–diffusion problems by the method of characteristics. SIAM J. Numer. Anal. 39, 858–879 (2001) · Zbl 1011.65064 · doi:10.1137/S0036142998336643
[28] Kačur, J., van Keer, R.: Solution of contaminant transport with adsorption in porous media by the method of characteristics. ESAIM M2AN Math. Model. Numer. Anal. 35, 981–1006 (2001) · Zbl 0995.76070 · doi:10.1051/m2an:2001146
[29] Knabner, P., Otto, F.: Solute transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: uniqueness of weak solutions. Nonlinear Anal. 42, 381–403 (2000) · Zbl 0958.35074 · doi:10.1016/S0362-546X(98)00352-6
[30] Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer. Math. 112, 115–152 (2009) · Zbl 1165.65063 · doi:10.1007/s00211-008-0203-5
[31] Manzini, A.C.G.: Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Methods Appl. Mech. Eng. 197(9–12), 933–945 (2008) · Zbl 1159.76356 · doi:10.1016/j.cma.2007.11.014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.