A superconvergent finite element method for the Korteweg-de Vries equation
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- by Douglas N. Arnold and Ragnar Winther PDF
- Math. Comp. 38 (1982), 23-36 Request permission
Abstract:
An unconditionally stable fully discrete finite element method for the Korteweg-de 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 time-independent 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.References
- D. N. Arnold and J. Douglas Jr., Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable, Calcolo 16 (1979), no. 4, 345–369 (1980). MR 592476, DOI 10.1007/BF02576636
- Douglas N. Arnold, Jim Douglas Jr., and Vidar Thomée, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), no. 153, 53–63. MR 595041, DOI 10.1090/S0025-5718-1981-0595041-4
- T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no. 1220, 47–78. MR 427868, DOI 10.1098/rsta.1972.0032
- J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
- Jim Douglas Jr., Todd Dupont, and Mary F. Wheeler, A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations, Math. Comp. 32 (1978), no. 142, 345–362. MR 495012, DOI 10.1090/S0025-5718-1978-0495012-2
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- Ragnar Winther, A conservative finite element method for the Korteweg-de Vries equation, Math. Comp. 34 (1980), no. 149, 23–43. MR 551289, DOI 10.1090/S0025-5718-1980-0551289-5
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 38 (1982), 23-36
- MSC: Primary 65M60; Secondary 76A60, 76B15
- DOI: https://doi.org/10.1090/S0025-5718-1982-0637284-8
- MathSciNet review: 637284