Convergence of a high order method in time and space for the miscible displacement equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 953-976.

A numerical method is formulated and analyzed for solving the miscible displacement problem under low regularity assumptions. The scheme employs discontinuous Galerkin time stepping with mixed and interior penalty discontinuous Galerkin finite elements in space. The numerical approximations of the pressure, velocity, and concentration converge to the weak solution as the mesh size and time step tend to zero. To pass to the limit a compactness theorem is developed which generalizes the Aubin−Lions theorem to accommodate discontinuous functions both in space and in time.

Reçu le :
DOI : 10.1051/m2an/2014059
Classification : 65M12, 65M60
Mots-clés : Generalized Aubin−Lions, discontinuous Galerkin, mixed finite element, arbitrary order, weak solution, convergence
Li, Jizhou 1 ; Riviere, Beatrice 1 ; Walkington, Noel 2

1 Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA
2 Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
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Li, Jizhou; Riviere, Beatrice; Walkington, Noel. Convergence of a high order method in time and space for the miscible displacement equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 953-976. doi : 10.1051/m2an/2014059. http://archive.numdam.org/articles/10.1051/m2an/2014059/

S. Bartels, M. Jensen and R. Müller, Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM J. Numer. Anal. 47 (2009) 3720–3743. | DOI | MR | Zbl

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. In vol. 15. Springer-Verlag (2007). | MR | Zbl

F. Brezzi, J. Douglas, M. Fortin and L. Marini, Efficient rectangular mixed finite elements in two and three space variables. RAIRO Model. Math. Anal. Numer. 21 (1987) 581–604. | DOI | Numdam | MR | Zbl

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Number 15 in Comput. Math. Springer-Verlag (1991). | MR | Zbl

A. Buffa and C. Ortner, Variational convergence of IP-DGFEM. Technical Report (2007).

A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29 (2009) 827–855. | DOI | MR | Zbl

Z. Chen and R.E. Ewing, Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30 (1999) 431–453. | DOI | MR | Zbl

D. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comput. 79 (2010) 1303–1330. | DOI | MR | Zbl

J. Douglas, R.E. Ewing and M.F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO. Numer. Anal. 17 (1983) 249–265. | DOI | Numdam | MR | Zbl

Y. Epshteyn and B.B. Rivière, Convergence of high order methods for miscible displacement. Int. J. Numer. Anal. Model. 5 (2008) 47–63. | MR | Zbl

K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Model. Math. Anal. Numer. 19 (1985) 611–643. | DOI | Numdam | MR | Zbl

R. Ewing and M.F. Wheeler, Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17 (1980) 351–365. | DOI | MR | Zbl

R.E. Ewing and T. Russell, Efficient time-stepping methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 19 (1982) 1–67. | DOI | MR | Zbl

X. Feng, On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194 (1995) 883–910. | DOI | MR | Zbl

M. Jensen and R. Müller, Stable Crank−Nicolson discretisation for incompressible miscible displacement problems of low regularity. Numer. Math. Adv. Appl. (2010) 469–477.

M. Ohlberger, Convergence of a mixed finite element – finite volume method for the two phase flow in porous media. East-Weat J. Numer. Math. 5 (1997) 183–210. | MR | Zbl

B. Riviere and N.J. Walkington, Convergence of a discontinuous Galerkin method for the miscible displacement under low regularity. SIAM J. Numer. Anal. 49 (2011) 1085—1110. | DOI | MR | Zbl

T.F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal. 22 (1985) 970–1013. | DOI | MR | Zbl

R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. American Mathematical Society, Providence, RI (1997). Available online at http://www.ams.org/online_bks/surv49/. | MR | Zbl

S. Sun, B. Riviere and M.F. Wheeler, A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media. Recent Progress in Computational and Applied PDEs (2002) 323–348. | MR | Zbl

N.J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations. SIAM J. Numer. Anal. 47 (2010) 4680–4170. | DOI | MR | Zbl

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