Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

L2-ERROR ANALYSIS OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR NONLINEAR SOBOLEV EQUATIONS

  • Ohm, Mi-Ray (Division of Information Systems Engineering Dongseo University) ;
  • Lee, Hyun-Young (Department of Mathematics Kyungsung University)
  • Received : 2010.10.01
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we develop a symmetric Galerkin method with interior penalty terms to construct fully discrete approximations of the solution for nonlinear Sobolev equations. To analyze the convergence of discontinuous Galerkin approximations, we introduce an appropriate projection and derive the optimal $L^2$ error estimates.

Keywords

References

  1. D. N. Arnold, An interior penalty nite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 724-760.
  2. D. N. Arnold, J. Jr. Douglas, and V. Thomee, Superconvergence of a nite approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), 53-63. https://doi.org/10.1090/S0025-5718-1981-0595041-4
  3. I. Babuska and M. Suri, The h-p version of the nite element method with quasi-uniform meshes, RAIRO Model. Math. Anal. Numer. 21 (1987), no. 2, 199-238. https://doi.org/10.1051/m2an/1987210201991
  4. I. Babuska and M. Suri, The optimal convergence rates of the p-version of the nite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750-776. https://doi.org/10.1137/0724049
  5. G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in ssured rocks, J. Appl. Math. Mech. 23 (1960), 1286-1303.
  6. R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Math-ematics in Sciences and Engineering, Vol. 127, Academic Press, New York, 1976.
  7. P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew Math. Phys. (1968), 614-627.
  8. P. L. Davis, A quasilinear parabolic and a related third order problem, J. Math. Anal. Appl. 40 (1972), 327-335. https://doi.org/10.1016/0022-247X(72)90054-6
  9. R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal. 6 (1975), 283-294. https://doi.org/10.1137/0506029
  10. R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equa- tions, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125-1150. https://doi.org/10.1137/0715075
  11. W. H. Ford and T. W. Ting, Stability and convergence of difference approximations to pseudo-parabolic partial differential equations, Math. Comput. 27 (1973), 737-743. https://doi.org/10.1090/S0025-5718-1973-0366052-4
  12. W. 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. https://doi.org/10.1137/0711016
  13. Y. Lin, Galerkin methods for nonlinear Sobolev equations, Aequationes Math. 40 (1990), no. 1, 54-66. https://doi.org/10.1007/BF02112280
  14. Y. Lin and T. Zhang, Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions, J. Math. Anal. Appl. 165 (1992), no. 1, 180-191. https://doi.org/10.1016/0022-247X(92)90074-N
  15. M. T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equa- tions in one space dimension, Numer. Math. 47 (1985), no. 1, 139-157. https://doi.org/10.1007/BF01389881
  16. M. R. Ohm, H. Y. Lee, and J. Y. Shin, Error estimates for discontinuous Galerkin method for nonlinear parabolic equations, J. Math. Anal. Appl. 315 (2006), no. 1, 132- 143. https://doi.org/10.1016/j.jmaa.2005.07.027
  17. M. R. Ohm, H. Y. Lee, and J. Y. Shin, $L^2$-error analysis of discontinuous Galerkin approximations for nonlinear Sobolev equations, submitted.
  18. T. Sun and D. Yang, A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations, Appl. Math. Comput. 200 (2008), no. 1, 147-159. https://doi.org/10.1016/j.amc.2007.10.053
  19. T. Sun and D. Yang, Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numer. Methods Partial Differential Equations 24 (2008), no. 3, 879-896. https://doi.org/10.1002/num.20294
  20. T. W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl. 45 (1974), 289-303.
  21. M. F. Wheeler, A priori $L_2$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723-759. https://doi.org/10.1137/0710062

Cited by

  1. A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED FINITE ELEMENT METHOD FOR THE CONVECTION DOMINATED SOBOLEV EQUATIONS vol.34, pp.1_2, 2016, https://doi.org/10.14317/jami.2016.019
  2. A CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS vol.32, pp.5, 2016, https://doi.org/10.7858/eamj.2016.051
  3. Fully discrete approximation of general nonlinear Sobolev equations pp.2190-7668, 2019, https://doi.org/10.1007/s13370-018-0626-9
  4. Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations pp.1572-9044, 2018, https://doi.org/10.1007/s10444-018-9628-2