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), 53-63

MSC:
Primary 65N30; Secondary 35K70

DOI:
https://doi.org/10.1090/S0025-5718-1981-0595041-4

MathSciNet review:
595041

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.

**[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. MR**0427868**, https://doi.org/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.)**[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. MR**0385355**, https://doi.org/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. MR**0400907****[5]**Richard E. Ewing,*Numerical solution of Sobolev partial differential equations*, SIAM J. Numer. Anal.**12**(1975), 345–363. MR**0395265**, https://doi.org/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. MR**512687**, https://doi.org/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. MR**0408270**, https://doi.org/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. MR**0366052**, https://doi.org/10.1090/S0025-5718-1973-0366052-4**[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. MR**0423833**, https://doi.org/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). MR**0287144**, https://doi.org/10.1002/mana.19700470133**[11]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[12]**J. L. Lions,*Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires*, Dunod, Paris, 1969.**[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. MR**0466924**, https://doi.org/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. MR**0454422**, https://doi.org/10.1137/0508062**[15]**Manuel Milla Miranda,*Weak solutions of a modified KdV equation*, Bol. Soc. Brasil. Mat.**6**(1975), no. 1, 57–63. MR**0467038**, https://doi.org/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. MR**0468211**, https://doi.org/10.1007/BF02584873**[18]**R. E. Showalter,*Sobolev equations for nonlinear dispersive systems*, Applicable Anal.**7**(1977/78), no. 4, 297–308. MR**504616**, https://doi.org/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.**[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. MR**0388799**, https://doi.org/10.1007/BF01438256

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
35K70

Retrieve articles in all journals with MSC: 65N30, 35K70

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0595041-4

Article copyright:
© Copyright 1981
American Mathematical Society