Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. (English) Zbl 1203.76078

Summary: This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beaver-Joseph-Saffman’s condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.


76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Adams, R.: Sobolev Spaces. Academic Press, Dordrecht (1975) · Zbl 0314.46030
[2] Arbogast, T., Brunson, D.: A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11(3), 207–218 (2007) · Zbl 1186.76660
[3] Badea, L., Discacciati, M., Quarteroni, A.: Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Milan (2006) · Zbl 1423.35304
[4] Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30, 197–207 (1967)
[5] Brenner, S.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Num. Anal. 41, 306–324 (2003) · Zbl 1045.65100
[6] Brenner, S.: Korn’s inequalities for piecewise h 1 vector fields. Math. Comput. 73, 1067–1087 (2004) · Zbl 1055.65118
[7] Burman, E., Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007) · Zbl 1101.76032
[8] Cesmelioglu, A., Rivière, B.: Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. (2008, to appear)
[9] Chidyagwai, P., Rivière, B.: On the solution of the coupled Navier-Stokes and Darcy equations (2007, submitted) · Zbl 1230.76023
[10] Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Eng. 193, 2565–2580 (2004) · Zbl 1067.76565
[11] Discacciati, M., Quarteroni, A.: Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. In: Brezzi, F., (eds.) Numerical Analysis and Advanced Applications–ENUMATH 2001, pp. 3–20. Springer, Berlin (2003) · Zbl 1254.76051
[12] Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2001) · Zbl 1023.76048
[13] Discacciati, M., Quarteroni, A., Valli, A.: Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45(3), 1246–1268 (2007) · Zbl 1139.76030
[14] Epshteyn, Y., Rivière, B.: Estimation of penalty parameters for symmetric interior penalty Galerkin methods. J. Comput. Appl. Math. 206, 843–872 (2007). doi: 10.1016/j.cam.2006.08.029 · Zbl 1141.65078
[15] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5. Springer, Berlin (1986) · Zbl 0585.65077
[16] Girault, V., Rivière, B.: DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. Technical Report TR-MATH 07-09, University of Pittsburgh (2007) · Zbl 1406.76082
[17] Girault, V., Rivière, B., Wheeler, M.: A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comput. 74, 53–84 (2004) · Zbl 1057.35029
[18] Girault, V., Rivière, B., Wheeler, M.: A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations. Math. Model. Numer. Anal. 39, 1115–1148 (2005) · Zbl 1085.76037
[19] Hanspal, N.S., Waghode, A.N., Nassehi, V., Wakeman, R.J.: Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media 64(1), 1573–1634 (2006) · Zbl 1309.76195
[20] Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000) · Zbl 0969.76088
[21] Mu, M., Xu, J.: A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45, 1801–1813 (2007) · Zbl 1146.76031
[22] Nassehi, V.: Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53(6), 1253–1265 (1998)
[23] Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22, 479–500 (2005) · Zbl 1065.76143
[24] Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42, 1959–1977 (2005) · Zbl 1084.35063
[25] Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput. Geosci. 3, 337–360 (1999) · Zbl 0951.65108
[26] Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902–931 (2001) · Zbl 1010.65045
[27] Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 292–315 (1971) · Zbl 0271.76080
[28] Salinger, A.G., Aris, R., Derby, J.J.: Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains. Int. J. Numer. Methods Fluids 18(1), 1185–1209 (1994) · Zbl 0807.76039
[29] Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40(6), 2171–2194 (2002) · Zbl 1055.76032
[30] Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978) · Zbl 0384.65058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.