Numerical methods for flows through porous media. I

Rose, Michael E.

doi:10.1090/S0025-5718-1983-0689465-6

Numerical methods for flows through porous media. I
HTML articles powered by AMS MathViewer

by Michael E. Rose PDF

Math. Comp. 40 (1983), 435-467 Request permission

Abstract:

The degenerate parabolic equation \[ \frac {{\partial u}}{{\partial t}} = \nabla \cdot \left ( {|u{|^\nu }\nabla u} \right ),\quad \nu \geqslant 1,\] has been used to model the flow of gas through a porous medium. Error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved, and several new regularity results are given.

References

D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461–467. MR 247303, DOI 10.1137/0117045
D. G. Aronson, Regularity properties of flows through porous media: The interface, Arch. Rational Mech. Anal. 37 (1970), 1–10. MR 255996, DOI 10.1007/BF00249496
D. G. Aronson, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math. 19 (1970), 299–307. MR 265774, DOI 10.1137/0119027

Equations d’Evolution dans un Espace de Banach Quelconque et Applications

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR 0348562
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Jim Douglas Jr., The flow of fluids in porous media, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 65–69. MR 0426607
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 371077, DOI 10.1090/S0025-5718-1975-0371077-0
Barry F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), no. 2, 381–415. MR 492856, DOI 10.1090/S0002-9947-1977-0492856-3

Linear and Quasilinear Equations of Parabolic Type

Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires

J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857
J. A. Nitsche, On $L_{\infty }$-convergence of finite element approximations to the solution of a nonlinear boundary value problem, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London-New York, 1977, pp. 317–325. MR 513215

On Some Degenerate Quasilinear Parabolic Equations

O. A. Oleĭnik, A. S. Kalašinkov, and Yuĭ-Lin′Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 667–704 (Russian). MR 0099834

Fundamentals of Numerical Reservoir Simulation

Numerical Methods for a General Class of Porous Medium Equations

Comput. Math. Appl.

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
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
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