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Li, Xindong; Rui, Hongxing

A rectangular mixed finite element method with a continuous flux for an elliptic equation modelling Darcy flow. (English) Zbl 1291.76207

Abstr. Appl. Anal. 2013, Article ID 580461, 10 p. (2013).
Summary: We introduce a mixed finite element method for an elliptic equation modelling Darcy flow in porous media. We use a staggered mesh where the two components of the velocity and the pressure are defined on three different sets of grid nodes. In the present mixed finite element, the approximate velocity is continuous and the conservation law still holds locally. The LBB consistent condition is established, while the \(L2\) error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to confirm the theoretical analysis.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
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\textit{X. Li} and \textit{H. Rui}, Abstr. Appl. Anal. 2013, Article ID 580461, 10 p. (2013; Zbl 1291.76207)
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References:

[1] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1977), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands
[2] Girault, V.; Raviart, P.-A., Finite Element Approximations of the Navier-Stokes Equations, 749 (1979), New York, NY, USA: Springer, New York, NY, USA · Zbl 0413.65081
[3] Brezzi, F.; Douglas,, J.; Durán, R.; Fortin, M., Mixed finite elements for second order elliptic problems in three variables, Numerische Mathematik, 51, 2, 237-250 (1987) · Zbl 0631.65107 · doi:10.1007/BF01396752
[4] Brezzi, F.; Douglas,, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, Mathematical Modelling and Numerical Analysis, 21, 4, 581-604 (1987) · Zbl 0689.65065
[5] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15 (1991), New York, NY, USA: Springer, New York, NY, USA · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[6] Raviart, P.-A.; Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the FEM. Mathematical Aspects of the FEM, Lecture Notes in Mathematics, 606, 292-315 (1977), Springer · Zbl 0362.65089
[7] Correa, M. R.; Loula, A. F. D., Unconditionally stable mixed finite element methods for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 197, 17-18, 1525-1540 (2008) · Zbl 1194.76109 · doi:10.1016/j.cma.2007.11.025
[8] Masud, A.; Hughes, T. J. R., A stabilized mixed finite element method for Darcy flow, Computer Methods in Applied Mechanics and Engineering, 191, 39-40, 4341-4370 (2002) · Zbl 1015.76047 · doi:10.1016/S0045-7825(02)00371-7
[9] Bear, J., Dynamics of Fluids in Porous Media (1972), New York, NY, USA: Dover, New York, NY, USA · Zbl 1191.76001
[10] Rui, H.; Tabata, M., A mass-conservative characteristic finite element scheme for convection-diffusion problems, Journal of Scientific Computing, 43, 3, 416-432 (2010) · Zbl 1203.65187 · doi:10.1007/s10915-009-9283-3
[11] Arbogast, T.; Wheeler, M. F., A family of rectangular mixed elements with a continuous flux for second order elliptic problems, SIAM Journal on Numerical Analysis, 42, 5, 1914-1931 (2005) · Zbl 1081.65106 · doi:10.1137/S0036142903435247
[12] Han, H. D., An economical finite element scheme for Navier-Stokes equations, Journal of Computational Mathematics, 5, 2, 135-143 (1987) · Zbl 0627.76031
[13] Han, H.; Wu, X., A new mixed finite element formulation and the MAC method for the Stokes equations, SIAM Journal on Numerical Analysis, 35, 2, 560-571 (1998) · Zbl 0918.76036 · doi:10.1137/S0036142996300385
[14] Han, H.; Yan, M., A mixed finite element method on a staggered mesh for Navier-Stokes equations, Journal of Computational Mathematics, 26, 6, 816-824 (2008) · Zbl 1174.35094
[15] Babuška, I., Error-bounds for finite element method, Numerische Mathematik, 16, 322-333 (1971) · Zbl 0214.42001
[16] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, 8, 2, 129-151 (1974) · Zbl 0338.90047
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