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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.