Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable
Authors:
Douglas N. Arnold, Jim Douglas and Vidar Thomée
Journal:
Math. Comp. 36 (1981), 5363
MSC:
Primary 65N30; Secondary 35K70
MathSciNet review:
595041
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Abstract: A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewisepolynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit singlestep methods is discussed.
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 [1]
 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)
 [2]
 J. L. Bona, W. G. Pritchard & L. R. Scott, "A comparison of laboratory experiments with a model equation for water waves." (To appear.)
 [3]
 J. L. Bona & R. Smith, "The initialvalue problem for the KortewegdeVries equation," Philos. Trans. Roy. Soc. London Ser. A, v. 278, 1975, pp. 555601. MR 0385355 (52:6219)
 [4]
 J. C. Eilbeck & G. R. McGuire, "Numerical studies of the regularized longwave equations I: numerical methods," J. Computational Phys., v. 19, 1975, pp. 4357. MR 0400907 (53:4737)
 [5]
 R. E. Ewing, "Numerical solution of Sobolev partial differential equations," SIAM J. Numer. Anal., v. 12, 1975, pp. 345363. MR 0395265 (52:16062)
 [6]
 R. E. Ewing, "Timestepping Galerkin methods for nonlinear Sobolev partial differential equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 11251150. MR 512687 (80b:65136)
 [7]
 W. H. Ford, "Galerkin approximations to nonlinear pseudoparabolic partial differential equations," Aequationes Math., v. 14, 1976, pp. 271291. MR 0408270 (53:12035)
 [8]
 W. H. Ford & T. W. Ting, "Stability and convergence of difference approximations to pseudoparabolic partial differential equations," Math. Comp., v. 27, 1973, pp. 737743. MR 0366052 (51:2303)
 [9]
 W. H. Ford & T. W. Ting, "Uniform error estimates for difference approximations to nonlinear pseudoparabolic partial differential equations," SIAM J. Numer. Anal., v. 11, 1974, pp. 155169. MR 0423833 (54:11807)
 [10]
 H. Gajewski & K. Zacharias, "Zur starken Konvergenz des Galerkinverfahrens bei einer Klasse pseudoparabolischer partieller Differentialgleichungen," Math. Nachr., v. 47, 1970, pp. 365376. MR 0287144 (44:4351)
 [11]
 P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 0135729 (24:B1772)
 [12]
 J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
 [13]
 L. A. Medeiros & M. Milla Miranda, "Weak solutions for a nonlinear dispersive equation," J. Math. Anal. Appl., v. 59, 1977, pp. 432441. MR 0466924 (57:6798)
 [14]
 L. A. Medeiros & G. Perla Menzala, "Existence and uniqueness for periodic solutions of the BenjaminBonaMahony equation," SIAM J. Math. Anal., v. 8, 1977, pp. 792799. MR 0454422 (56:12673)
 [15]
 M. Milla Miranda, "Weak solutions of a modified KdV equation," Bol. Soc. Brasil. Mat., v. 6, 1975, pp. 5763. MR 0467038 (57:6907)
 [16]
 D. H. Peregrine, "Calculations of the development of an undular bore," J. Fluid Mech., v. 25, 1966, pp. 321330.
 [17]
 M. A. Raupp, "Galerkin methods applied to the BenjaminBonaMahony equation," Bol. Soc. Brasil. Mat., v. 6, 1975, pp. 6577. MR 0468211 (57:8049)
 [18]
 R. E. Showalter, "Sobolev equations for nonlinear dispersive systems," Applicable Anal., v. 7, 1978, pp. 297308. MR 504616 (80g:34067)
 [19]
 R. E. Showalter & T. W. Ting, "Pseudoparabolic partial differential equations," SIAM J. Math. Anal., v. 1, 1970, pp. 126.
 [20]
 L. Wahlbin, "Error estimates for a Galerkin method for a class of model equations for long waves," Numer. Math., v. 23, 1975, pp. 289303. MR 0388799 (52:9633)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198105950414
PII:
S 00255718(1981)05950414
Article copyright:
© Copyright 1981
American Mathematical Society
