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Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable. (English) Zbl 0466.65062


MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI

References:

[1] 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. · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[2] J. L. Bona, W. G. Pritchard & L. R. Scott, ”A comparison of laboratory experiments with a model equation for water waves.” (To appear.) · Zbl 0497.76023
[3] 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. · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035
[4] J. C. Eilbeck and G. R. McGuire, Numerical study of the regularized long-wave equation. I. Numerical methods, J. Computational Phys. 19 (1975), no. 1, 43 – 57. · Zbl 0325.65054
[5] Richard E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM J. Numer. Anal. 12 (1975), 345 – 363. · Zbl 0355.65071 · doi:10.1137/0712028
[6] Richard E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125 – 1150. · Zbl 0399.65083 · doi:10.1137/0715075
[7] William H. Ford, Galerkin approximations to non-linear pseudo-parabolic partial differential equations, Aequationes Math. 14 (1976), no. 3, 271 – 291. · Zbl 0343.65047 · doi:10.1007/BF01835978
[8] William H. Ford and T. W. Ting, Stability and convergence of difference approximations to pseudo-parabolic partial differential equations, Math. Comp. 27 (1973), 737 – 743. · Zbl 0271.65053
[9] William H. Ford and T. W. Ting, Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations, SIAM J. Numer. Anal. 11 (1974), 155 – 169. · Zbl 0244.65064 · doi:10.1137/0711016
[10] Herbert Gajewski and Klaus Zacharias, Zur starken Konvergenz des Galerkinverfahrens bei einer Klasse pseudoparabolischer partieller Differentialgleichungen, Math. Nachr. 47 (1970), 365 – 376 (German). · Zbl 0217.53102 · doi:10.1002/mana.19700470133
[11] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. · Zbl 0112.34901
[12] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. · Zbl 0189.40603
[13] L. A. Medeiros and M. Milla Miranda, Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl. 59 (1977), no. 3, 432 – 441. · Zbl 0376.35011 · doi:10.1016/0022-247X(77)90071-3
[14] L. A. Medeiros and Gustavo Perla Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal. 8 (1977), no. 5, 792 – 799. · Zbl 0337.35004 · doi:10.1137/0508062
[15] Manuel Milla Miranda, Weak solutions of a modified KdV equation, Bol. Soc. Brasil. Mat. 6 (1975), no. 1, 57 – 63. · Zbl 0383.35008 · doi:10.1007/BF02584872
[16] D. H. Peregrine, ”Calculations of the development of an undular bore,” J. Fluid Mech., v. 25, 1966, pp. 321-330.
[17] M. A. Raupp, Galerkin methods applied to the Benjamin-Bona-Mahony equation, Bol. Soc. Brasil. Mat. 6 (1975), no. 1, 65 – 77. · Zbl 0385.65051 · doi:10.1007/BF02584873
[18] R. E. Showalter, Sobolev equations for nonlinear dispersive systems, Applicable Anal. 7 (1977/78), no. 4, 297 – 308. · Zbl 0387.34043 · doi:10.1080/00036817808839200
[19] R. E. Showalter & T. W. Ting, ”Pseudo-parabolic partial differential equations,” SIAM J. Math. Anal., v. 1, 1970, pp. 1-26. · Zbl 0199.42102
[20] Lars Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975), 289 – 303. With an appendix by Lars Wahlbin, Jim Douglas, Jr. and Todd Dupont. · Zbl 0283.65052 · doi:10.1007/BF01438256
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