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Error estimates for the finite volume discretization for the porous medium equation. (English) Zbl 1402.65093

Summary: We analyze the convergence of a numerical scheme for a class of degenerate parabolic problems modelling reactions in porous media, and involving a nonlinear, possibly vanishing diffusion. The scheme involves the Kirchhoff transformation of the regularized nonlinearity, as well as an Euler implicit time stepping and triangle based finite volumes. We prove the convergence of the approach by giving error estimates in terms of the discretization and regularization parameter.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
35K57 Reaction-diffusion equations
35K65 Degenerate parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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[1] Cariaga, E.; Concha, F.; Sepúlveda, M., Flow through porous media with applications to heap leaching of copper ores, Chem. Eng. J., 111, 151-165 (2005)
[2] Cariaga, E.; Concha, F.; Sepúlveda, M., Convergence of a MFE-FV method for two phase flow with applications to heap leaching of copper ores, Comput. Methods Appl. Mech. Engrg., 196, 2541-2554 (2007) · Zbl 1173.76333
[3] Ohlberger, M., Convergence of a mixed finite element - finite volume method for the two phase flow in porous media, East-West J. Numer. Math., 5, 183-210 (1997) · Zbl 0899.76261
[4] Nochetto, R. H.; Verdi, C., Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25, 784-814 (1988) · Zbl 0655.65131
[5] Arbogast, T.; Wheeler, M. F.; Zhang, N. Y., 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
[6] Radu, F. A.; Pop, I. S.; Knabner, P., Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal., 42, 1452-1478 (2004) · Zbl 1159.76352
[7] Radu, F. A.; Pop, I. S.; Knabner, P., Error estimates for a mixed finite element discretization of some degenerate parabolic equations, Numer. Math., 109, 285-311 (2008) · Zbl 1141.65071
[8] Schneid, E.; Knabner, P.; Radu, F. A., A priori error estimates for a mixed finite element discretization of the Richards’ equation, Numer. Math., 98, 353-370 (2004) · Zbl 1075.76042
[9] Riviere, B.; Wheeler, M. F., Discontinuous Galerkin methods for flow and transport problems in porous media, Comm. Numer. Methods Engrg., 18, 63-68 (2002) · Zbl 0996.76056
[10] Klausen, R. A.; Radu, F. A.; Eigestad, G. T., Convergence of MPFA on triangulations and for Richards’ equation, Internat. J. Numer. Methods Fluids, 58, 1327-1351 (2008) · Zbl 1391.76341
[11] Afif, M.; Amaziane, B., Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in flow in porous media, Comput. Methods Appl. Mech. Engrg., 191, 5265-5286 (2002) · Zbl 1012.76057
[12] 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
[13] 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
[14] Alt, H. W.; Luckhaus, S., Quasilinear elliptic-parabolic differential equations, Math. Z., 183, 311-341 (1983) · Zbl 0497.35049
[15] Otto, F., \(L^1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations, 131, 20-38 (1996) · Zbl 0862.35078
[16] Jäger, W.; Kačur, J., Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, \(M^2\) AN (Math. Model. Numer. Anal.), 29, 605-627 (1995) · Zbl 0837.65103
[18] Pop, I. S., Error estimates for a time discretization method for the Richards’ equation, Comput. Geosci., 6, 141-160 (2002) · Zbl 1079.65542
[19] Pop, I. S.; Yong, W. A., A numerical approach to degenerate parabolic equations, Numer. Math., 92, 357-381 (2002) · Zbl 1011.65063
[20] Pop, I. S., Numerical schemes for degenerate parabolic problems, (Bucchianico, A. Di; Mattheij, R. M.M.; Peletier, M. A., Progress in Industrial Mathematics at ECMI 2004. Progress in Industrial Mathematics at ECMI 2004, Mathematics in Industry, Vol. 8 (2006), Springer-Verlag: Springer-Verlag Heidelberg), 513-517 · Zbl 1309.76196
[21] Rulla, J., Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. Numer. Anal., 33, 68-87 (1996) · Zbl 0855.65102
[22] Eymard, R.; Gallouët, T.; Herbin, R., Finite volume methods, (Handbook of Numerical Analysis, Vol. VII (2000), North-Holland: North-Holland Amsterdam), 713-1020 · Zbl 0981.65095
[23] Gallouët, T.; Herbin, R.; Vignal, M. H., Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions, SIAM J. Numer. Anal., 37, 1935-1972 (2000) · Zbl 0986.65099
[24] Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (2001), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI · Zbl 0981.35001
[25] Coudiére, Y.; Gallouët, T.; Herbin, R., Discrete Sobolev inequalities and Lp error estimates for approximate finite volume solutions of convection diffusion equations, \(M^2\) AN (Math. Model. Numer. Anal.), 35, 767-778 (2001) · Zbl 0990.65122
[26] Baranger, J.; Maître, J. F.; Oudin, F., Connection between finite volume and mixed finite element methods, RAIRO - Modél. Math. Anal. Numér., 30, 445-465 (1996) · Zbl 0857.65116
[27] Chen, Z., Expanded mixed finite element methods for quasilinear second order elliptic problems, \(M^2\) AN (Math. Model. Numer. Anal.), 32, 501-520 (1998) · Zbl 0910.65080
[28] Herbin, R., An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, Numer. Methods Partial Differential Equations, 11, 165-173 (1995) · Zbl 0822.65085
[29] Kröner, D., Numerical Schemes for Conservation Laws (1997), Wiley-Teubner: Wiley-Teubner Stuttgart · Zbl 0872.76001
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