Computational issues of hybrid and multipoint mixed methods for groundwater flow in anisotropic media. (English) Zbl 1398.76104

Summary: In this work, lowest-order Raviart-Thomas and Brezzi-Douglas-Marini mixed methods are considered for groundwater flow simulations. Typically, mixed methods lead to a saddle-point problem, which is expensive to solve. Two approaches are numerically compared here to allow an explicit velocity elimination: (1) the well-known hybrid formulation leading to a symmetric positive definite system where the only unknowns are the Lagrange multipliers and (2) a more recent approach, inspired from the multipoint flux approximation method, reducing low-order mixed methods to cell-centered finite difference schemes. Selected groundwater flow scenarios are used for the comparison between hybrid and multipoint approaches. The simulations are performed in the bidimensional case with a general triangular discretization because of its practical interest for hydrogeologists.


76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[1] Aavatsmark, I.: An introduction to multi-point flux approximation for quadrilateral grids. J. Comput. Geosci. 6(3–4), 405–432 (2002) · Zbl 1094.76550
[2] Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127, 2–14 (1996) · Zbl 0859.76048
[3] Aavatsmark, I., Eigestad, T., Klausen, R.A., Wheeler, M.F., Yotov, I.: Convergence of a symmetric MPFA method on quadrilateral grids. J. Comput. Geosci. 11(4), 333–345 (2007) · Zbl 1128.65093
[4] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15. Springer, New York (1991) · Zbl 0788.73002
[5] Brezzi, F., Fortin, M., Marini, L.D.: Error analysis of piecewise constant approximations of Darcy’s law. Comput. Methods Appl. Mech. Eng. 195(13–16), 1547–1599 (2006) · Zbl 1116.76051
[6] Chavent, G., Roberts, J.E.: A unified physical presentation of mixed, mixed-hybrid finite elements and usual finite differences for the determination of velocities in waterflow problems. Adv. Water Resour. 14(6), 323–352 (1991)
[7] Davis, T.A., Duff, I.S.: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25(1), 1–20 (1999) · Zbl 0962.65027
[8] Eigestad, G.T., Klausen, R.A.: Convergence of the MPFA O-method: numerical experiments for discontinuous media. J. Numer. Methods Partial Differ. Equ. 21, 1079–1098 (2005) · Zbl 1089.76037
[9] Eisenstat, S.C.: Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM J. Sci. Statist. Comput. 2, 1–4 (1981) · Zbl 0474.65020
[10] Fontaine, V., Younès, A.: On the multipoint mixed finite volume methods on quadrilateral grids. In: Proceedings of the XVI international conference on computational methods in water resources, p. 10 (2006)
[11] Fraeijs de Veubeke, B.X.: Displacement and Equilibrium Models in the Finite Element Method. Stress Analysis, New York (1965) · Zbl 0359.73007
[12] Klausen, R.A., Russell, T.F.: Relationships among some locally conservative discretization methods which handle discontinuous coefficients. J. Comput. Geosci. 8, 341–377 (2004) · Zbl 1124.76030
[13] Klausen, R.A., Winther, R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104, 317–337 (2006) · Zbl 1102.76036
[14] Leij, F.J., Dane, J.H.: Analytical solution of the one-dimensional advection equation and two or three-dimensional dispersion equation. Water Resour. Res. 26, 1475–1482 (1990)
[15] Raviart, P.A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. In: Mahematical Aspects of Finite Element Method. Lecture Notes in Mathematics, no. 606, pp. 292–315. Springer, New York (1977)
[16] Roberts, J.E., Thomas, J.-M.: Mixed and hybrid finite element methods. In: Handbook of Numerical Analysis. Finite Element Methods, vol. II, Part. 1, pp. 523–639. North-Holland, Amsterdam (1991) · Zbl 0875.65090
[17] Vohralík, M.: Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. M2AN Math. Model. Numer. Anal. 40(2), 367–391 (2006) · Zbl 1116.65121
[18] Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44(5), 2082–2106 (2006) · Zbl 1121.76040
[19] Younès, A., Ackerer, P., Chavent, G.: From mixed finite elements to finite volumes for elliptic PDEs in 2 and 3 dimensions. Int. J. Numer. Methods Eng. 59(3), 365–388 (2004) · Zbl 1043.65131
[20] Younès, A., Fontaine, V.: Efficiency of mixed hybrid finite element and multipoint flux approximations methods on quadrangular grids and highly anisotropic media. Int. J. Numer. Methods Eng. 76(3), 314–336 (2008) · Zbl 1195.74208
[21] Younès, A., Fontaine, V.: Hybrid and multi point formulations of the lowest order mixed methods for darcy’s flow on triangles. Int. J. Numer. Methods Fluids 58(9), 1041–1062 (2008) · Zbl 1149.76030
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