A stable extrapolation method for multidimensional degenerate parabolic problems
Author:
Ricardo H. Nochetto
Journal:
Math. Comp. 53 (1989), 455470
MSC:
Primary 65N30; Secondary 35K55, 35K65, 65N15
MathSciNet review:
982372
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Abstract: Degenerate parabolic problems in several space variables are approximated by combining a preliminary regularization procedure with a finite element extrapolation method. The proposed extrapolation acts on the socalled phase variable and leads to a linear problem which is shown to be stable. The ensuing linear algebraic system involves the same matrix for all time steps. Energy error estimates are also derived for the physical unknowns. An rate of convergence is proved, provided the approximation parameters are suitably related. In case the linear systems are solved by an iterative algorithm, such as the conjugate gradient method, an tolerance for the error reduction is shown to preserve the overall accuracy; the required computational effort is thus nearly optimal.
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 [1]
 O. Axelsson & V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Applications, Academic Press, Orlando, FL, 1984. MR 758437 (85m:65116)
 [2]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [3]
 J. Douglas, Jr. & T. Dupont, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 0277126 (43:2863)
 [4]
 J. Douglas, Jr. & T. Dupont, "Alternatingdirection Galerkin methods on rectangles," in Numerical Solutions of Partial Differential Equations, vol. II (B. Hubbard, ed.), Academic Press, New York, 1971, pp. 133214. MR 0273830 (42:8706)
 [5]
 J. Douglas, Jr. & T. Dupont, "Galerkin methods for parabolic equations with nonlinear boundary conditions," Numer. Math., v. 20, 1973, pp. 213217. MR 0319379 (47:7923)
 [6]
 J. Douglas, Jr., T. Dupont & R. Ewing, "Incomplete iteration for timestepping a Galerkin method for a quasilinear parabolic problem," SIAM J. Numer. Anal., v. 16, 1979, pp. 503522. MR 530483 (80f:65117)
 [7]
 A. Friedman, "The Stefan problem in several space variables," Trans. Amer. Math. Soc., v. 133, 1968, pp. 5187. MR 0227625 (37:3209)
 [8]
 J. W. Jerome & M. Rose, "Error estimates for the multidimensional twophase Stefan problem," Math. Comp., v. 39, 1982, pp. 377414. MR 669635 (84h:65097)
 [9]
 M. Luskin, "A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions," SIAM J. Numer. Anal., v. 16, 1979, pp. 284299. MR 526490 (80f:65121)
 [10]
 E. Magenes, "Problemi di Stefan bifase in più variabili spaziali," Matematiche, v. 36, 1981, pp. 65108. MR 736797 (85f:35198)
 [11]
 E. Magenes, "Remarques sur l'approximation des problèmes non linéaires paraboliques," in Analyse Mathématique et Applications (volume dedicated to J.L. Lions), GauthierVillars, Paris, 1988, pp. 297318. MR 956965 (90f:65158)
 [12]
 E. Magenes, R. H. Nochetto & C. Verdi, "Energy error estimates for a linear scheme to approximate nonlinear parabolic problems," RAIRO Modél. Math. Anal. Numér., v. 21, 1987, pp. 655678. MR 921832 (89b:65220)
 [13]
 R. H. Nochetto, "Error estimates for twophase Stefan problems in several space variables, I: Linear boundary conditions," Calcolo, v. 22, 1985, pp. 457499. MR 859087 (88a:65122a)
 [14]
 R. H. Nochetto, "Error estimates for multidimensional singular parabolic problems," Japan. J. Appl. Math., v. 4, 1987, pp. 111138. MR 899207 (89c:65107)
 [15]
 R. H. Nochetto, "Numerical methods for free boundary problems," in Free Boundary Problems: Theory and Applications (K. H. Hoffmann and J. Sprekels, eds.), vols. V, VI, Research Notes in Math., Longman, London, 1988. (To appear.)
 [16]
 R. H. Nochetto & C. Verdi, "An efficient linear scheme to approximate parabolic free boundary problems: Error estimates and implementation," Math. Comp., v. 51, 1988, pp. 2753. MR 942142 (89k:65124)
 [17]
 R. H. Nochetto & C. Verdi, "The combined use of a nonlinear Chernoff formula with a regularization procedure for twophase Stefan problems," Numer. Funct. Anal. Optim., v. 9, 198788, pp. 11771192. MR 936337 (89b:65205)
 [18]
 M. Paolini, G. Sacchi & C. Verdi, "Finite element approximations of singular parabolic problems," Internat. J. Numer. Methods Engrg., v. 26, 1988, pp. 19892007. MR 955582 (89j:76023)
 [19]
 M. Rose, "Numerical methods for flows through porous media, I," Math. Comp., v. 40, 1983, pp. 435467. MR 689465 (85a:65146)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909823726
PII:
S 00255718(1989)09823726
Keywords:
Mushy region,
regularization,
extrapolation,
finite elements
Article copyright:
© Copyright 1989
American Mathematical Society
