Abstract
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.
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Adams, R.: Sobolev Spaces. Academic Press, Dordrecht (1975)
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)
Badea, L., Discacciati, M., Quarteroni, A.: Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Milan (2006)
Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30, 197–207 (1967)
Brenner, S.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Num. Anal. 41, 306–324 (2003)
Brenner, S.: Korn’s inequalities for piecewise h 1 vector fields. Math. Comput. 73, 1067–1087 (2004)
Burman, E., Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007)
Cesmelioglu, A., Rivière, B.: Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. (2008, to appear)
Chidyagwai, P., Rivière, B.: On the solution of the coupled Navier-Stokes and Darcy equations (2007, submitted)
Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Eng. 193, 2565–2580 (2004)
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)
Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2001)
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)
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
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5. Springer, Berlin (1986)
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)
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)
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)
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)
Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)
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)
Nassehi, V.: Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53(6), 1253–1265 (1998)
Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22, 479–500 (2005)
Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42, 1959–1977 (2005)
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)
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)
Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 292–315 (1971)
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)
Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40(6), 2171–2194 (2002)
Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)
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The authors acknowledge the support of NSF through the grants DMS 0506039 and DMS 0810422.
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Çeşmelioğlu, A., Rivière, B. Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow. J Sci Comput 40, 115–140 (2009). https://doi.org/10.1007/s10915-009-9274-4
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DOI: https://doi.org/10.1007/s10915-009-9274-4