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Unconditionally stable mixed finite element methods for Darcy flow. (English) Zbl 1194.76109
Summary: Unconditionally stable finite element methods for Darcy flow are derived by adding least squares residual forms of the governing equations to the classical mixed formulations. The proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous Lagrangian finite element spaces of any order for the velocity and the potential. Stability, convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence.
MSC:
76M10Finite element methods (fluid mechanics)
76S05Flows in porous media; filtration; seepage
65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
References:
[1]Raviart, P. A.; Thomas, J. M.: A mixed finite element method for second order elliptic problems, , 292-315 (1977) · Zbl 0362.65089
[2]Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods, Springer series in computational mathematics 15 (1991) · Zbl 0788.73002
[3]Loula, A. F. D.; Rochinha, F. A.; Murad, M. A.: Higher-order gradient post-processings for second-order elliptic problems, Comput. methods appl. Mech. engrg. 128, 361-381 (1995) · Zbl 0862.65061 · doi:10.1016/0045-7825(95)00885-3
[4]Correa, M. R.; Loula, A. F. D.: Stabilized velocity post-processings for Darcy flow in heterogenous porous media, Commun. numer. Methods engrg. 23, 461-489 (2007)
[5]Cordes, C.; Kinzelbach, W.: Continuous groundwater velocity fields and path lines in linear, bilinear and trilinear finite elements, Water resources res. 28, No. 11, 2903-2911 (1992)
[6]Durlofsky, L. J.: Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water resources res. 30, No. 4, 965-973 (1994)
[7]Brezzi, F.; Existence, On The: Uniqueness and approximation of saddle point problems arising from Lagrange multipliers, Analyse numérique/numerical analysis (RAIRO) 8, No. R-2, 129-151 (1974) · Zbl 0338.90047
[8]Loula, A. F. D.; Hughes, T. J. R.; Franca, L. P.: Mixed Petrov – Galerkin methods for the Timoshenko beam problem, Comput. methods appl. Mech. engrg. 63, 133-154 (1987)
[9]A.F.D. Loula, E.M. Toledo, Dual and primal mixed Petrov – Galerkin finite element methods in heat transfer problems, LNCC-Technical Report 048/88, 1988.
[10]Franca, L. P.; Hughes, T. J. R.; Loula, A. F. D.; Miranda, I.: A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov – Galerkin method, Numer. math. 53, 123-141 (1988) · Zbl 0656.73036 · doi:10.1007/BF01395881
[11]Masud, A.; Hughes, T. J. R.: A stabilized finite element method for Darcy flow, Comput. methods appl. Mech. engrg. 191, 4341-4370 (2002) · Zbl 1015.76047 · doi:10.1016/S0045-7825(02)00371-7
[12]Brezzi, F.; Hughes, T. J. R.; Marini, L. D.; Masud, A.: A mixed discontinuous Galerkin method for Darcy flow, SIAM J. Sci. comput. 22 – 23, 119-145 (2005) · Zbl 1103.76031 · doi:10.1007/s10915-004-4150-8
[13]Hughes, T. J. R.; Masud, A.; Wan, J.: A stabilized mixed discontinuous Galerkin method for Darcy flow, Comput. methods appl. Mech. engrg. 195, 3347-3381 (2006) · Zbl 1120.76040 · doi:10.1016/j.cma.2005.06.018
[14]Barrenechea, G.; Franca, L. P.; Valentin, F.: A Petrov Galerkin enriched method: a mass conservative finite element method for the Darcy equation, Comput. methods appl. Mech. engrg. 196, 2449-2464 (2007) · Zbl 1173.76329 · doi:10.1016/j.cma.2007.01.004
[15]Ciarlet, P. G.: The finite element method for elliptic problems, Studies in mathematics and its applications (1978)
[16]Zienkiewicz, O. C.; Zhu, J. Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique, Int. J. Numer. methods engrg. 33, 1331-1364 (1992) · Zbl 0769.73084 · doi:10.1002/nme.1620330702
[17]Carey, G. F.; Oden, J. T.: Finite elements: mathematical aspects, Finite elements: mathematical aspects 4 (1983)
[18]Girault, V.; Raviart, P.: Finite element methods for Navier – Stokes equations: theory and algorithms, Springer series in computational mathematics (1986)
[19]Brezzi, F.; Jr., J. Douglas; Marini, L. D.: Two families of mixed finite elements for second order elliptic problems, Numer. math. 47, 217-235 (1985) · Zbl 0599.65072 · doi:10.1007/BF01389710
[20]Cai, Z.; Manteuffel, T. A.; Mccormick, S. F.: First-order system least squares for second-order partial differential equations: part II, SIAM J. Numer. anal. 34, No. 2, 425-454 (1997) · Zbl 0912.65089 · doi:10.1137/S0036142994266066
[21]Babu&caron, I.; Ska: Error bounds for finite element method, Numer. math. 16, 133-322 (1971)
[22]Gelfand, I. M.; Fomin, S. V.: Calculus of variations, (1963)
[23]Nakshatrala, K. B.; Turner, D. Z.; Hjelmstad, K. D.; Masud, A.: A stabilized finite element method for Darcy flow based on a multiscale decomposition of the solution, Comput. methods appl. Mech. engrg. 195, 4036-4049 (2006) · Zbl 1125.76044 · doi:10.1016/j.cma.2005.07.009
[24]A.F.D. Loula, M.R. Correa, Numerical analysis of stabilized mixed finite element methods for Darcy flow, in: III European Conference on Computational Mechanics – ECCM 2006, Lisbon, Portugal, 2006.
[25]Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations, Analyse numérique/numerical analysis (RAIRO) 18, No. 2, 175-182 (1984)
[26]M.R. Correa, Stabilized finite element methods for Darcy and coupled Stokes – Darcy Flows, D.Sc. Thesis, LNCC, Petrópolis, RJ, Brazil, 2006 (in Portuguese).
[27]Discacciati, M.; Miglio, E.; Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows, Appl. numer. Math. 43, 57-74 (2002) · Zbl 1023.76048 · doi:10.1016/S0168-9274(02)00125-3
[28]Layton, W. J.; Schieweck, F.; Yotov, I.: Coupling fluid flow with porous media flow, SIAM J. Numer. anal. 40, No. 6, 2195-2218 (2003) · Zbl 1037.76014 · doi:10.1137/S0036142901392766
[29]Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. sci. Comput. 22 – 23, 479-500 (2005) · Zbl 1065.76143 · doi:10.1007/s10915-004-4147-3
[30]Arbogast, T.; Brunson, D. S.: A computational method for approximating a Darcy – Stokes system governing a vuggy porous medium, Comput. geosci. 11, No. 3, 207-218 (2007) · Zbl 1186.76660 · doi:10.1007/s10596-007-9043-0
[31]M.R. Correa, A. Loula, An adjoint stabilized mixed finite element method for porous media flow, in: XXVIII CILAMCE – Iberian Latin – American Congress on Computational Methods in Engineering, Porto, Portugal, 2007 (in Portuguese).
[32]Jr., J. Douglas; Wang, J.: An absolutely stabilized finite element method for the Stokes problem, Math. comput. 52, No. 186, 495-508 (1989) · Zbl 0669.76051 · doi:10.2307/2008478
[33]Morel-Seytoux, H. J.: Analytical numerical method in water flooding predictions, Spe j. 5, 247-285 (1965)