Mixed finite element methods for compressible miscible displacement in porous media

Chou, So-Hsiang; Li, Qian

doi:10.1090/S0025-5718-1991-1094942-7

Mixed finite element methods for compressible miscible displacement in porous media
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by So-Hsiang Chou and Qian Li PDF

Math. Comp. 57 (1991), 507-527 Request permission

Abstract:

A differential system describing compressible miscible displacement in a porous medium is given. The concentration equation is treated by a Galerkin method and the pressure equation is treated by a parabolic mixed finite element method. Optimal-order estimates in ${L^2}$ and almost optimal-order estimates in ${L^\infty }$ are obtained for the errors in the approximate solutions under the condition that $h_p^{2k + 2}{(\log h_c^{ - 1})^{1/2}} \to 0$. This condition is much weaker than one given earlier by Douglas and Roberts for the same model. Furthermore, we obtain the ${L^\infty }({L^2}(\Omega ))$-estimates for the time-derivatives of the concentration and the pressure, which were not given by the above authors. In addition, we also consider newer mixed spaces in two or three dimensions.

References

Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581–604 (English, with French summary). MR 921828, DOI 10.1051/m2an/1987210405811
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Jim Douglas Jr. and Jean E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp. 41 (1983), no. 164, 441–459. MR 717695, DOI 10.1090/S0025-5718-1983-0717695-3

Optimal

estimates for some Galerkin methods for the Dirichlet problem

Claes Johnson and Vidar Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér. 15 (1981), no. 1, 41–78 (English, with French summary). MR 610597
J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
A. H. Schatz, V. C. Thomée, and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math. 33 (1980), no. 3, 265–304. MR 562737, DOI 10.1002/cpa.3160330305
Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI 10.1090/S0025-5718-1976-0436617-2
Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI 10.1137/0710062