A superconvergent finite element method for the KortewegdeVries equation
Authors:
Douglas N. Arnold and Ragnar Winther
Journal:
Math. Comp. 38 (1982), 2336
MSC:
Primary 65M60; Secondary 76A60, 76B15
MathSciNet review:
637284
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Abstract: An unconditionally stable fully discrete finite element method for the Kortewegde Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small timeindependent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.
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 D. N. Arnold & J. Douglas, Jr., ``Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable,'' Calcolo, v. 16, 1979, pp. 345369. MR 592476 (83b:65118)
 [2]
 D. N. Arnold, J. Douglas, Jr. & V. Thomée, ``Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable,'' Math. Comp., v. 36, 1981, pp. 5363. MR 595041 (82f:65108)
 [3]
 T. B. Benjamin, J. L. Bona & J. J. Mahony, ``Model equations for long waves in nonlinear dispersive systems,'' Philos. Trans. Roy. Soc. London Ser. A, v. 272, 1972, pp. 4778. MR 0427868 (55:898)
 [4]
 J. L. Bona & R. Smith, ``The initialvalue problem for the Kortewegde Vries equation,'' Philos. Trans. Roy. Soc. London Ser. A, v. 278, 1975, pp. 555601. MR 0385355 (52:6219)
 [5]
 J. Douglas, Jr., T. Dupont & M. F. Wheeler, ``A quasiprojection analysis of Galerkin methods for parabolic and hyperbolic equations,'' Math. Comp., v. 32, 1978, pp. 345362. MR 0495012 (58:13780)
 [6]
 G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)
 [7]
 R. Winther, ``A conservative finite element method for the Kortewegde Vries equation,'' Math. Comp., v. 34, 1980, pp. 2343. MR 551289 (81a:65108)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206372848
PII:
S 00255718(1982)06372848
Article copyright:
© Copyright 1982 American Mathematical Society
