East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED ELEMENT METHOD FOR SOBOLEV EQUATIONS WITH A CONVECTION TERM

  • Ohm, Mi Ray (Division of Mechatronics Engineering Dongseo University) ;
  • Shin, Jun Yong (Department of Applied Mathematics Pukyong National University)
  • Received : 2019.08.24
  • Accepted : 2019.09.26
  • Published : 2019.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider a split least-squares characteristic mixed element method for Sobolev equations with a convection term. First, to manipulate both convection term and time derivative term efficiently, we apply a characteristic mixed element method to get the system of equations in the primal unknown and the flux unknown and then get a least-squares minimization problem and a least-squares characteristic mixed element scheme. Finally, we obtain a split least-squares characteristic mixed element scheme for the given problem whose system is uncoupled in the unknowns. We prove the optimal order in $L^2$ and $H^1$ normed spaces for the primal unknown and the suboptimal order in $L^2$ normed space for the flux unknown.

Keywords

Acknowledgement

Supported by : Pukyong National University

References

  1. T. Arbogast and M. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problem, SIAM J. Numer. Anal. 32(2) (1995), 404-424. https://doi.org/10.1137/0732017
  2. G. I. Barenblatt, I. P. Zheltov and I. N. Kochian, Basic conception in the theory of seepage of homogenous liquids in fissured rocks, J. Appl. Math. Mech. 24 (1960), 1286-1309. https://doi.org/10.1016/0021-8928(60)90107-6
  3. K. Boukir, Y. Maday and B. Metivet, A high-order characteristics/finite element method for the incompressible navier-stokes equations, Inter. Jour. Numer. Methods in Fluids. 25 (1997), 1421-1454. https://doi.org/10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A
  4. Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems, Comput. Methods. Appl. Mech. Engrg. 191 (2002), 2509-2538. https://doi.org/10.1016/S0045-7825(01)00411-X
  5. Z. Chen, R. Ewing, Q. Jiang and A. Spagnuolo, Error analysis for characteristics-based methods for degenerate parabolic problems, SIAM J. Numer. Anal. 40(4) (2002), 1491-1515. https://doi.org/10.1137/S003614290037068X
  6. C. Dawson, T. Russell and M. Wheeler, Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal. 26(6) (1989), 1487-1512. https://doi.org/10.1137/0726087
  7. J. Douglas and T. F. Russell Jr., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristic with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), 871-885. https://doi.org/10.1137/0719063
  8. R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev equations, SIAM J. Numer. Anal. 15 (1978), 1125-1150. https://doi.org/10.1137/0715075
  9. 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), 341-351. https://doi.org/10.1016/j.matcom.2009.07.003
  10. H. Gu, Characteristic finite element methods for nonlinear Sobolev equations, Applied Math. Compu. 102 (1999), 51-62. https://doi.org/10.1016/S0096-3003(98)10019-X
  11. L. Guo and H. Z. Chen, $H^1$-Galerkin mixed finite element method for the Sobolev equation, J. Sys. Sci. 26 (2006), 301-314.
  12. H. Guo and H. X. Rui, Least-squares Galerkin mixed finite element method for the Sobolev equation, Acta Math. Appl. Sinica 29 (2006), 609-618. https://doi.org/10.3321/j.issn:0254-3079.2006.04.004
  13. X. Long and C. Chen, Implicit-Explicit multistep characteristic finite element methods for nonlinear convection-diffusion equations, Numer. Methods Parial Differential Eq. 23 (2007), 1321-1342. https://doi.org/10.1002/num.20222
  14. M. R. Ohm and H. Y. Lee, $L^2$-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations, Bull. Korean. Math. Soc. 48(5) (2011), 897-915. https://doi.org/10.4134/BKMS.2011.48.5.897
  15. M. R. Ohm, H. Y. Lee and J. Y. Shin, $L^2$-error analysis of discontinuous Galerkin approximations for nonlinear Sobolev equations, J. Japanese Indus. Appl. Math. 30(1) (2013), 91-110. https://doi.org/10.1007/s13160-012-0096-7
  16. M. R. Ohm and J. Y. Shin, A split least-squares characteristics mixed finite element method for the convection dominated Sobolev equations, J. Appl. Math. Informatics. 34(1) (2016), 19-34. https://doi.org/10.14317/jami.2016.019
  17. M. R. Ohm and J. Y. Shin, A Crank-Nicolson characteristic finite element method for Sobolev equations, East Asian Math. J. 30(5) (2016), 729-744.
  18. A. Pehlivanov, G. F. Carey and D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994), 1368-1377. https://doi.org/10.1137/0731071
  19. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathemaical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin, 1977, 292-315.
  20. H. X. Rui, S. Kim and S. D. Kim, A remark on least-squares mixed element methods for reaction-diffusion problems, J. Comput. Appl. Math. 202 (2007), 203-236. https://doi.org/10.1016/j.cam.2006.02.025
  21. D. M. Shi, On the initial boundary value problem of the nonlinear equation of the migration of the moisture in soil, Acta math. Appl. Sinica 13 (1990), 31-38.
  22. 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
  23. D. P. Yang, Some least-squares Galerkin procedures for first-order time-dependent convection-diffusion system, Comput. Methods Appl. Mech. Eng. 108 (1999), 81-95. https://doi.org/10.1016/S0045-7825(99)00050-X
  24. D. P. Yang, Analysis of least-squares mixed finite element methods for nonlinear non-stationary convection-diffusion problems, Math. Comput. 69 (2000), 929-963. https://doi.org/10.1090/S0025-5718-99-01172-2
  25. J. Zhang amd H. Guo, A split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem, Int. J. Comput. Math. 89 (2012), 932-943. https://doi.org/10.1080/00207160.2012.667086