A PRIORI ERROR ESTIMATES OF A DISCONTINUOUS GALERKIN METHOD FOR LINEAR SOBOLEV EQUATIONS

  • Ohm, Mi-Ray (DIVISION OF INFORMATION SYSTEMS ENGINEERING, DONGSEO UNIVERSITY) ;
  • Shin, Jun-Yong (DIVISION OF MATHEMATICAL SCIENCES, PUKYONG NATIONAL UNIVERSITY) ;
  • Lee, Hyun-Young (DEPARTMENT OF MATHEMATICS, KYUNGSUNG UNIVERSITY)
  • Received : 2009.03.03
  • Accepted : 2009.06.16
  • Published : 2009.09.25

Abstract

A discontinuous Galerkin method with interior penalty terms is presented for linear Sobolev equation. On appropriate finite element spaces, we apply a symmetric interior penalty Galerkin method to formulate semidiscrete approximate solutions. To deal with a damping term $\nabla{\cdot}({\nabla}u_t)$ included in Sobolev equations, which is the distinct character compared to parabolic differential equations, we choose special test functions. A priori error estimate for the semidiscrete time scheme is analyzed and an optimal $L^\infty(L^2)$ error estimation is derived.

Acknowledgement

Supported by : Kyungsung University

References

  1. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), 724–760.
  2. I. Babuska, M. Suri, The h-p version of the finite element method with quasi-uniform meshes, RAIRO Model. Math. Anal. Numer. 21 (1987), 199–238. https://doi.org/10.1051/m2an/1987210201991
  3. I. Babuska, M. Suri, The optimal convergence rates of the p-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), 750–776. https://doi.org/10.1137/0724049
  4. G. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), 45–59. https://doi.org/10.1090/S0025-5718-1977-0431742-5
  5. J. Douglas, T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lect. Notes. Phys. 58 (1976), 207–216.
  6. J. A. Nitsche, Uber ein Variationspringzip zur Losung von Dirichlet-Problemen bei Verwendung von Teiliaumen, die Keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15. https://doi.org/10.1007/BF02995904
  7. J. T. Oden, I. Babuska, C. E. Baumann, A discontinuous hp finite element method for diffusion provlems, J. Comput. Phys. 146 (1998), 491–519. https://doi.org/10.1006/jcph.1998.6032
  8. M. R. Ohm, H. Y. Lee, J. Y. Shin, Error estimates for discontinuous Galerkin method for nonlinear parabolic equations, Journal of Math. Anal. and Appli., 315 (2006), 132–143. https://doi.org/10.1016/j.jmaa.2005.07.027
  9. B. Riviere, M. F. Wheeler, K. Banas, Part II. Discontinuous Galerkin method applied to single phase flow in porous media, Comput. Geosci. 4(4) (2000), 337–341. https://doi.org/10.1023/A:1011546411957
  10. B. Riviere, M. F. Wheeler, V. Girault, Part I. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, Comput. Geosci. 8 (1999), 337–360.
  11. B. Riviere, M. F. Wheeler, V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39(3) (2001), 902–931. https://doi.org/10.1137/S003614290037174X
  12. T. Sun, D. Yang, Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numerical Methods Partial Differential Equations 24(3) (2008), 879–896. https://doi.org/10.1002/num.20294
  13. T. Sun, D. Yang, A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations, Applied Mathematics and Computation 200 (2008), 147–159. https://doi.org/10.1016/j.amc.2007.10.053
  14. M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), 152–161. https://doi.org/10.1137/0715010