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Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow

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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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References

  1. Adams, R.: Sobolev Spaces. Academic Press, Dordrecht (1975)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Badea, L., Discacciati, M., Quarteroni, A.: Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Milan (2006)

  4. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30, 197–207 (1967)

    Article  Google Scholar 

  5. Brenner, S.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Num. Anal. 41, 306–324 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brenner, S.: Korn’s inequalities for piecewise h 1 vector fields. Math. Comput. 73, 1067–1087 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Burman, E., Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  10. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Eng. 193, 2565–2580 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  12. Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  15. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5. Springer, Berlin (1986)

    MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  20. Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nassehi, V.: Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53(6), 1253–1265 (1998)

    Article  Google Scholar 

  23. Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22, 479–500 (2005)

    Article  MathSciNet  Google Scholar 

  24. Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42, 1959–1977 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 292–315 (1971)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  29. Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40(6), 2171–2194 (2002)

    Article  MathSciNet  Google Scholar 

  30. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6100 Main Street MS-134, Houston, TX, 77005, USA

    Ayçıl Çeşmelioğlu & Béatrice Rivière

Authors
  1. Ayçıl Çeşmelioğlu
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  2. Béatrice Rivière
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Correspondence to Béatrice Rivière.

Additional information

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

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