Browse > Article
http://dx.doi.org/10.7858/eamj.2015.049

ERROR ESTIMATES FOR FULLY DISCRETE MIXED DISCONTINUOUS GALERKIN APPROXIMATIONS FOR PARABOLIC PROBLEMS  

OHM, MI RAY (DIVISION OF MECHATRONICS ENGINEERING, DONGSEO UNIVERSITY)
LEE, HYUN YOUNG (DEPARTMENT OF MATHEMATICS, KYUNGSUNG UNIVERSITY)
SHIN, JUN YONG (DEPARTMENT OF APPLIED MATHEMATICS, PUKYONG NATIONAL UNIVERSITY)
Publication Information
East Asian mathematical journal / v.31, no.5, 2015 , pp. 685-693 More about this Journal
Abstract
In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.
Keywords
Convection dominated Sobolev equations; a split least squares method; characteristic mixed finite element method; convergence of optimal order;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), 724 - 760.
2 D. N. Arnold, F. Breezi, B. Cockburn, L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001), 1749 - 1779.
3 Z. Chen, On the relationship of various discontinuous finite element methods for secondorder elliptic equations, East-West J. Numer. Math. 9 (2001), 99 - 122.
4 H. Chen, Z. Chen, Stability and convergence of mixed discontinuous finite element methods for second-order diffferential problems, J. Numer. Math. 11 (2003), 253 - 287.   DOI   ScienceOn
5 Z. Chen, J. Douglas, Approximation of coecients in hybrid and mixed methods for nonlinear parabolic problems, Math. Applic. Comp. 10 (1991), 137 - 160.
6 J. Douglas, T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lect. Notes. Phys. 58 (1976), 207 - 216.   DOI
7 J. Douglas, J. E. Roberts, Global estimates for mixed methods for second order elliptic problems, Math. Comp. 44 (1985), 39 - 52.   DOI   ScienceOn
8 I. Guo, H. Z. Chen, $H^1$-Galerkin mixed finite element methods for the Sobolev equation, Systems Sci. Math. Sci. 26 (2006), 301 - 314.
9 C. Johnson, V. Thomee, Error estimates for som mixed finite element methods for parabolic type problems, RAIRO Anal. Numer. 14 (1981), 41 - 78.
10 J. -C. Nedelec, Mixed finite elements in $\mathbb{R}^3$, Numer. Math. 35 (1980), 315 - 341.   DOI
11 J. Nitsche, Uber ein Variationspringzip zvr Losung von Dirichlet-Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9 - 15.   DOI
12 M. R. Ohm, H. Y. Lee, J. Y. Shin, Error estimates for discontinuous Galerkin method for nonlinear parabolic equations, J. Math. Anal. Appl. 315 (2006), 132 - 143.   DOI   ScienceOn
13 A. K. Pani, An $H^1$-Galerkin mixed finite element methods for parabolic partial differential equations, SIAM J. Numer. Anal. 25 (1998), 712 - 727.
14 M. R. Ohm, H. Y. Lee, J. Y. Shin, Error estimates for fully discrete discontinuous Galerkin method for nonlinear parabolic equations, J. Appl. Math. & Informatics 28 (2010), 953 - 966.
15 M. R. Ohm, H. Y. Lee, J. Y. Shin, Higher order discontinuous Galerkin finite element methods for nonlinear parabolic equations, J. Korean Soc. Ind. Appl. Math. 18 (2014), 337 - 350.   DOI   ScienceOn
16 M. R. Ohm, J. Y. Shin, Error estimates for a semi-discrete mixed discontinuous Galerkin method with an interior penalty for parabolic problems, J. Korean Soc. Ind. Appl. Math. (submitted).
17 A. K. Pani, G. Fairweather, $H^1$-Galerkin mixed finite element methods for parabolic integro-differential equations, IMA J. Numer. Anal. 22 (2002), 231-252.   DOI   ScienceOn
18 R. Raviart, J. Thomas, A mixed finite element method for second order elliptic problems, Lecture Notes in Mathematics 606 (1977), 292 - 315.   DOI
19 B. Rivire, M. F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations, Discontinuous Galerkin methods:theory, computation and applications [Eds by B. Cockburn, G. E. Karniadakis and C. -W. Shu], Lect. Notes Comput. Sci. Eng. 11 (2000), 231 - 244.
20 M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), 152 - 161.   DOI   ScienceOn