### Electronic Research Archive

2021, Issue 2: 1991-2006. doi: 10.3934/era.2020101
Special Issues

# Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems

• Received: 01 April 2020 Revised: 01 July 2020 Published: 23 September 2020
• Primary: 65M60, 65N22

• We investigate the matrix structure of the discrete system of the multiscale discontinuous Galerkin method (MDG) for general second order partial differential equations [10]. The MDG solution is obtained by composition of DG and the inter-scale operator. We show that the MDG matrix is given by the product of the DG matrix and the inter-scale matrix of the local problem. We apply an ILU preconditioned GMRES to solve the matrix equation effectively. Numerical examples are presented for convection dominated problems.

Citation: ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems[J]. Electronic Research Archive, 2021, 29(2): 1991-2006. doi: 10.3934/era.2020101

### Related Papers:

• We investigate the matrix structure of the discrete system of the multiscale discontinuous Galerkin method (MDG) for general second order partial differential equations [10]. The MDG solution is obtained by composition of DG and the inter-scale operator. We show that the MDG matrix is given by the product of the DG matrix and the inter-scale matrix of the local problem. We apply an ILU preconditioned GMRES to solve the matrix equation effectively. Numerical examples are presented for convection dominated problems.

 [1] Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. (2001/02) 39: 1749-1779. [2] A multiscale discontinuous Galerkin method. Large-Scale Scientific Computing, Lecture Notes in Comput. Sci (2006) 3743: 84-93. [3] Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. (2006) 44: 1420-1440. [4] A sub-grid structure enhanced discontinuous galerkin method for multiscale diffusion and convection-diffusion problems. Commun. Comput. Phys. (2013) 14: 370-392. [5] Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. (2002) 39: 2133-2163. [6] A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Comput. Methods Appl. Mech. Engrg. (2006) 195: 2761-2787. [7] S. J. Jeong, A multiscale discontinuous Galerkin method for convection-diffusion-reaction problems: A numberical study, PhD Thesis. [8] hp-discontinuous Galerkin methods for the Lotka-McKendrick equation$:$ A numerical study. Commun. Korean Math. Soc. (2007) 22: 623-640. [9] A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Engrg. (2008) 197: 806-820. [10] A multiscale discontinuous Galerkin methods for convection-diffusion-reaction problems. Comput. Math. Appl. (2014) 68: 2251-2261. [11] Y. Saad, Iterative Methods for Sparse Linear Systems, 2$^nd$ edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003 [12] GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. (1986) 7: 856-869. [13] Ch. Schwab, p- and hp-finite element methods, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8

Article outline

## Figures and Tables

Figures(13)  /  Tables(4)

• On This Site