A superconvergent finite element method for the Korteweg-deVries equation

Authors:
Douglas N. Arnold and Ragnar Winther

Journal:
Math. Comp. **38** (1982), 23-36

MSC:
Primary 65M60; Secondary 76A60, 76B15

MathSciNet review:
637284

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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.

**[1]**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**, 10.1007/BF02576636**[2]**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**, 10.1090/S0025-5718-1981-0595041-4**[3]**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**0427868****[4]**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**0385355****[5]**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**0495012**, 10.1090/S0025-5718-1978-0495012-2**[6]**G. B. Whitham,*Linear and nonlinear waves*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0483954****[7]**Ragnar Winther,*A conservative finite element method for the Korteweg-de Vries equation*, Math. Comp.**34**(1980), no. 149, 23–43. MR**551289**, 10.1090/S0025-5718-1980-0551289-5

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0637284-8

Article copyright:
© Copyright 1982
American Mathematical Society