Central local discontinuous galerkin methods on overlapping cells for diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, p. 1009-1032

In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.

DOI : https://doi.org/10.1051/m2an/2011007
Classification:  65M60
Keywords: central discontinuous Galerkin method, local discontinuous Galerkin method, overlapping cells, diffusion equation, heat equation, stability, error estimate
@article{M2AN_2011__45_6_1009_0,
     author = {Liu, Yingjie and Shu, Chi-Wang and Tadmor, Eitan and Zhang, Mengping},
     title = {Central local discontinuous galerkin methods on overlapping cells for diffusion equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {6},
     year = {2011},
     pages = {1009-1032},
     doi = {10.1051/m2an/2011007},
     zbl = {1269.65098},
     mrnumber = {2833171},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_6_1009_0}
}
Liu, Yingjie; Shu, Chi-Wang; Tadmor, Eitan; Zhang, Mengping. Central local discontinuous galerkin methods on overlapping cells for diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, pp. 1009-1032. doi : 10.1051/m2an/2011007. http://www.numdam.org/item/M2AN_2011__45_6_1009_0/

[1] R. Bellman, The stability of solutions of linear differential equations. Duke Math. J. 10 (1943) 643-647. | MR 9408 | Zbl 0061.18502

[2] P. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland (1975). | MR 520174 | Zbl 0383.65058

[3] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463. | MR 1655854 | Zbl 0927.65118

[4] B. Cockburn and C.-W. Shu, Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173-261. | MR 1873283 | Zbl 1065.76135

[5] B. Cockburn, B. Dong, J. Guzman, M. Restelli and R. Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31 (2009) 3827-3846. | MR 2556564 | Zbl 1200.65093

[6] Y.J. Liu, C.-W. Shu, E. Tadmor and M. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45 (2007) 2442-2467. | MR 2361897 | Zbl 1157.65450

[7] Y.-J. Liu, C.-W. Shu, E. Tadmor and M. Zhang, L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: M2AN 42 (2008) 593-607. | Numdam | MR 2437775 | Zbl 1152.65095

[8] B. Van Leer and S. Nomura, Discontinuous Galerkin for diffusion, in Proceedings of 17th AIAA Computational Fluid Dynamics Conference (2005) 2005-5108.

[9] M. Van Raalte and B. Van Leer, Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Comm. Comput. Phys. 5 (2009) 683-693. | MR 2513709

[10] M. Zhang and C.-W. Shu, An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13 (2003) 395-413. | MR 1977633 | Zbl 1050.65094

[11] M. Zhang and C.-W. Shu, An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids 34 (2005) 581-592. | Zbl 1138.76391