DOI QR코드

DOI QR Code

A PRIORI L2 ERROR ANALYSIS FOR AN EXPANDED MIXED FINITE ELEMENT METHOD FOR QUASILINEAR PSEUDO-PARABOLIC EQUATIONS

  • Ohm, Mi Ray (Division of Information Systems Engineering Dongseo University) ;
  • Lee, Hyun Young (Department of Mathematics Kyungsung University) ;
  • Shin, Jun Yong (Department of Applied Mathematics Pukyong National University)
  • 투고 : 2012.09.18
  • 발행 : 2014.01.01

초록

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on ${\Omega}{\subset}R^d$, $1{\leq}d{\leq}3$. We construct the semidiscrete approximations of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u in $L^2$ normed space.

키워드

과제정보

연구 과제 주관 기관 : Dongseo University

참고문헌

  1. D. N. Arnold, J. Jr. Douglas, and V. Thomee, 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. https://doi.org/10.1090/S0025-5718-1981-0595041-4
  2. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 724-760.
  3. G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech. 24 (1960), 1286-1303. https://doi.org/10.1016/0021-8928(60)90107-6
  4. F. Brezzi, J. Jr. Douglas, and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217-235. https://doi.org/10.1007/BF01389710
  5. Y. Cao, J. Yin, and C.Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations 246 (2009), no. 12, 4568-4590. https://doi.org/10.1016/j.jde.2009.03.021
  6. R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Mathematics 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. 19 (1968), no. 4, 614-627. https://doi.org/10.1007/BF01594969
  8. Y. Chen and L. Li, $L^p$ error estimates of two-grid schemes of expanded mixed finite element methods, Appl. Math. Comp. 209 (2009), no. 2, 197-205. https://doi.org/10.1016/j.amc.2008.12.033
  9. P. L. Davis, A quasilinear parabolic and related third order problem, J. Math. Anal. Appl. 49 (1972), 327-335.
  10. J. Jr. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. https://doi.org/10.1090/S0025-5718-1985-0771029-9
  11. R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125-1150. https://doi.org/10.1137/0715075
  12. F. Gao and H. Rui, A split least-squares characteristic mixed finite element method for sobolev equations with convection term, Math. Comput. Simulation 80 (2009), no. 2, 341-351. https://doi.org/10.1016/j.matcom.2009.07.003
  13. F. Gao, J. Qiu, and Q. Zhang, Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation, J. Sci. Comput. 41 (2009), no. 3, 436-460. https://doi.org/10.1007/s10915-009-9308-y
  14. H. Guo, A remark on split least-squares mixed element procedures for pseudo-parabolic equations, Appl. Math. Comput. 217 (2011), no. 9, 4682-4690. https://doi.org/10.1016/j.amc.2010.11.021
  15. D. Kim and E.-J. Park, A posteriori error estimator for expanded mixed hybrid methods, Numer. Methods Partial Differential Equations 23 (2007), no. 2, 330-349. https://doi.org/10.1002/num.20178
  16. Y. Lin, Galerkin methods for nonlinear Sobolev equations, Aequationes Math. 40 (1990), no. 1, 54-66. https://doi.org/10.1007/BF02112280
  17. 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
  18. M. T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numer. Math. 47 (1985), no. 1, 139-157. https://doi.org/10.1007/BF01389881
  19. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.
  20. D. Shi and H. Wang, Nonconforming $H^1$-Galerkin mixed FEM for Sobolev equations on anisotropic meshes, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 2, 335-344. https://doi.org/10.1007/s10255-007-7065-y
  21. D. Shi and Y. Zhang, High accuracy analysis of a new nonconforming mixed finite element scheme for sobolev equation, Appl. Math. Comput. 218 (2011), no. 7, 3176-3186. https://doi.org/10.1016/j.amc.2011.08.054
  22. 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
  23. 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
  24. T. W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl. 45 (1974), 23-31. https://doi.org/10.1016/0022-247X(74)90116-4