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

ShinJa Jeong; Mi-Young Kim; ShinJa Jeong; Mi-Young Kim

doi:10.3934/era.2020101

Electronic Research Archive

2021, Volume 29, Issue 2: 1991-2006. doi: 10.3934/era.2020101

Previous Article Next Article

Special Issues

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

Department of Mathematics, Inha University, Incheon 22212, Korea

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.
- MDG,
- multiscale,
- convection-diffusion,
- DG
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:

Abstract

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.

References

[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.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)