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A stabilized mixed finite element method for Darcy flow. (English) Zbl 1015.76047

Summary: We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[1] Babuska, I., Error bounds for finite element methods, Numer. Math., 16, 322-333 (1971) · Zbl 0214.42001
[2] Bergamaschi, L.; Mantica, S.; Manzini, G., A mixed finite element-finite volume formulation of the black oil model, SIAM J. Sci. Comput., 20, 970-997 (1998) · Zbl 0959.76039
[3] Brezzi, F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O. Anal. Numer., 8, 129-151 (1974) · Zbl 0338.90047
[4] Brezzi, F.; Douglas, J.; Duran, R.; Fortin, M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51, 237-250 (1987) · Zbl 0631.65107
[5] Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, Math. Model. Numer. Anal., 21, 581-604 (1987) · Zbl 0689.65065
[6] Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235 (1985) · Zbl 0599.65072
[7] Brezzi, F.; Douglas, J.; Marini, L. D., Recent results on mixed finite elements for second order elliptic problems, (Balakrishanan; Dorodnitsyn; Lions, Vistas in Applied Math., Numerical Analysis, Atmospheric Sciences, Immunology (1986), Optimization Software Publications: Optimization Software Publications New York) · Zbl 0611.65071
[8] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15 (1991), Springer: Springer New York
[9] Brezzi, F.; Franca, L. P.; Hughes, T. J.R.; Russo, A., \(b\)=∫\(g\), Comput. Meth. Appl. Mech. Engrg., 145, 329-339 (1997) · Zbl 0904.76041
[10] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes equations, (Hackbush, W., Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, vol. 10 (1984), Braunschweig: Braunschweig Wiesbaden) · Zbl 0552.76002
[11] Brezzi, F.; Russo, A., Choosing bubbles for advection-diffusion problems, Math. Models Meth. Appl. Sci., 4, 571-587 (1994) · Zbl 0819.65128
[12] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin methods for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[13] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Meth. Fluid Mech. (1988), Springer: Springer Berlin
[14] Chavent, G.; Cohen, G.; Jaffre, J., Discontinuous upwinding and mixed finite elements for two-phase flows in reservoir simulation, Comput. Meth. Appl. Mech. Engrg., 47, 93-118 (1984) · Zbl 0545.76130
[15] J. Douglas Jr., F. Pereira, L.-M. Yeh, A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media, preprint, November 3, 2000; J. Douglas Jr., F. Pereira, L.-M. Yeh, A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media, preprint, November 3, 2000 · Zbl 0969.76069
[16] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52, 495-508 (1989) · Zbl 0669.76051
[17] Durlofsky, L. J., Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water Resour. Res., 30, 965-973 (1994)
[18] Ewing, R. E.; Heinemann, R. F., Mixed finite element approximation of phase velocities in compositional reservoir simulation, Comput. Meth. Appl. Mech. Engrg., 47, 161-175 (1984) · Zbl 0545.76127
[19] Ewing, R. E.; Russell, T. F.; Wheeler, M. F., Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Meth. Appl. Mech. Engrg., 47, 73-92 (1984) · Zbl 0545.76131
[20] Franca, L. P.; Frey, S. L.; Hughes, T. J.R., Stabilized finite element methods. I. Application to the advective-diffusive model, Comput. Meth. Appl. Mech. Engrg., 95, 253-276 (1992) · Zbl 0759.76040
[21] Franca, L. P.; Hughes, T. J.R., Two classes of mixed finite element methods, Comput. Meth. Appl. Mech. Engrg., 69, 89-129 (1988) · Zbl 0651.65078
[22] Franca, L. P.; Hughes, T. J.R.; Stenberg, S., Stabilized finite element methods for Stokes problem, (Gunzburger, M. D.; Nicolaides, R. A., Incompressible Computational Fluid Dynamics (1993), Cambridge University Press: Cambridge University Press Cambridge), 87-107 · Zbl 1189.76339
[23] L.P. Franca, A. Russo, A formulation based on residual-free bubbles for a singular perturbation equation, preprint; L.P. Franca, A. Russo, A formulation based on residual-free bubbles for a singular perturbation equation, preprint · Zbl 1043.65123
[24] Franca, L. P.; Russo, A., Unlocking with residual-free bubbles, Comput. Meth. Appl. Mech. Engrg., 142, 361-364 (1997) · Zbl 0890.73064
[25] Franca, L. P.; Stenberg, R., Error analysis for some Galerkin-least-squares methods for the elasticity equations, SIAM J. Numer. Anal., 28, 6, 1680-1797 (1991) · Zbl 0759.73055
[26] Hughes, T. J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Meth. Appl. Mech. Engrg., 127, 387-401 (1995) · Zbl 0866.76044
[27] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall: Prentice-Hall Englewoods Cliffs, NJ, (Dover edition, 2000) · Zbl 0634.73056
[28] Hughes, T. J.R.; Brezzi, F., On drilling degrees of freedom, Comput. Meth. Appl. Mech. Engrg., 72, 105-121 (1989) · Zbl 0691.73015
[29] Hughes, T. J.R.; Brooks, A. N., A multidimensional upwind scheme with no crosswind diffusion, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows (1979), ASME: ASME New York), 19-35 · Zbl 0423.76067
[30] Hughes, T. J.R.; Brooks, A. N., Streamline-upwind/Petrov-Galerkin methods for advection dominated flows, (Proceedings of the Third International Conference on Finite Element Methods in Fluid Flow, Banff (June 1980)), 283-292
[31] Hughes, T. J.R.; Brooks, A. N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: applications to the streamline upwind procedure, (Gallagher, R. H.; Carey, G. F.; Oden, J. T.; Zienkiewicz, O. C., Finite Elements in Fluids, vol. IV (1982), Wiley: Wiley Chichester), 46-65
[32] Hughes, T. J.R.; Brooks, A. N., An algorithm for solving the Navier-Stokes equations based upon the streamline-upwind/Petrov-Galerkin formulation, (Lewis, R. W.; etal., Numerical Methods in Coupled Systems (1984), Wiley: Wiley London), 387-404 · Zbl 0569.76036
[33] Hughes, T. J.R.; Engel, G.; Mazzei, L.; Larson, M. G., The continuous Galerkin method is locally conservative, J. Computat. Phys., 163, 467-488 (2000) · Zbl 0969.65104
[34] Hughes, T. J.R.; Feijoo, G.; Mazzei, L.; Quincy, J.-B., The variational multiscale method: a paradigm for computational mechanics, Comput. Meth. Appl. Mech. Engrg., 166, 3-24 (1998) · Zbl 1017.65525
[35] Hughes, T. J.R.; Franca, L. P., A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well posed boundary condition: symmetric formulations that converge for all velocity/pressure spaces, Comput. Meth. Appl. Mech. Engrg., 65, 86-96 (1987) · Zbl 0635.76067
[36] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations, Comput. Meth. Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077
[37] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Meth. Appl. Mech. Engrg., 73, 173-189 (1989) · Zbl 0697.76100
[38] Hughes, T. J.R.; Liu, W. K.; Brooks, A., Finite element analysis of incompressible viscous flows by the penalty function, J. Computat. Phys., 30, 1, 1-60 (1979) · Zbl 0412.76023
[39] Johnson, C., Streamline diffusion methods for problems in fluid mechanics, (Gallagher, R. H.; Carey, G. F.; Oden, J. T.; Zienkiewicz, O. C., Finite Elements In Fluids, Vol. VI (1986), Wiley: Wiley Chichester), 251-261
[40] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Meth. Appl. Mech. Engrg., 45, 285-312 (1984) · Zbl 0526.76087
[41] Loula, A. F.D.; Garcia, E. L.M.; Coutinho, A. L.G. A., Miscible displacement simulation by finite element methods in distributed memory machines, Comput. Meth. Appl. Mech. Engrg., 174, 354-361 (1999) · Zbl 0961.76044
[42] Loula, A. F.D.; Rochinha, F. A.; Murad, M. A., Higher-order gradient post-processing for second-order elliptic problems, Comput. Meth. Appl. Mech. Engrg., 128, 361-381 (1995) · Zbl 0862.65061
[43] Malkus, D. S.; Hughes, T. J.R., Mixed finite element methods-reduced and selective integration techniques: a unification of concepts, Comput. Meth. Appl. Mech. Engrg., 15, 63-81 (1978) · Zbl 0381.73075
[44] Malta, S. M.C.; Loula, A. F.D.; Garcia, E. L.M., Numerical analysis of stabilized finite element method for tracer injection simulations, Comput. Meth. Appl. Mech. Engrg., 187, 119-136 (2000) · Zbl 0958.76044
[45] Nedelec, J. C., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069
[46] Nedelec, J. C., A new family of mixed finite elements in \(R^3\), Numer. Math., 50, 57-81 (1986) · Zbl 0625.65107
[47] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer: Springer Berlin · Zbl 0852.76051
[48] Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606 (1977), Springer: Springer New York) · Zbl 0362.65089
[49] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations (1996), Springer: Springer Heidelberg
[50] J.M. Thomas, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, These d’Etat, Université Pierre et Marie Curie, Paris, 1977; J.M. Thomas, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, These d’Etat, Université Pierre et Marie Curie, Paris, 1977
[51] Wang, H.; Dahle, H. K.; Ewing, R. E.; Espedal, M. S.; Sharpley, R. D.; Man, S., An ELLAM scheme for advection-diffusion equations in two dimensions, SIAM J. Sci. Comput., 20, 2160-2194 (1999) · Zbl 0939.65109
[52] Wang, H.; Liang, D.; Ewing, R. E.; Lyons, S. L.; Qin, G., An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite elements, SIAM J. Sci. Comput., 22, 561-581 (2000) · Zbl 0988.76054
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