American Institute of Mathematical Sciences

Advanced
doi: 10.3934/era.2020101

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

ShinJa Jeong and  Mi-Young Kim

Department of Mathematics, Inha University, Incheon 22212, Korea

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: The second author was supported by INHA University Research Grant

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.

Keywords: MDG, multiscale, convection-diffusion, DG.
Mathematics Subject Classification: Primary: 65M60, 65N22.
Citation: ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, doi: 10.3934/era.2020101
References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[2]

P. Bochev, T. J. R. Hughes and G. Scovazzi, A multiscale discontinuous Galerkin method, Large-Scale Scientific Computing, Lecture Notes in Comput. Sci, 3743 (2006), 84-93.  doi: 10.1007/11666806_8.  Google Scholar

[3]

A. Buffa, T. J. R. Hughes and G. Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44 (2006), 1420-1440.  doi: 10.1137/050640382.  Google Scholar

[4]

E. T. Chung and W. T. Leung, A sub-grid structure enhanced discontinuous galerkin method for multiscale diffusion and convection-diffusion problems, Commun. Comput. Phys., 14 (2013), 370-392.  doi: 10.4208/cicp.071211.070912a.  Google Scholar

[5]

P. Houston, C. Schwab and E. Süli, Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.  doi: 10.1137/S0036142900374111.  Google Scholar

[6]

T. J. R. Hughes, G. Scovazzi, P. B. Bochev and A. Buffa, A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2761-2787.  doi: 10.1016/j.cma.2005.06.006.  Google Scholar

[7]

S. J. Jeong, A multiscale discontinuous Galerkin method for convection-diffusion-reaction problems: A numberical study, PhD Thesis. Google Scholar

[8]

S.-J. Jeong, M.-Y. Kim and T. Selenge, hp-discontinuous Galerkin methods for the Lotka-McKendrick equation$:$ A numerical study, Commun. Korean Math. Soc., 22 (2007), 623-640.  doi: 10.4134/CKMS.2007.22.4.623.  Google Scholar

[9]

D. Kim and E.-J. Park, A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems, Comput. Methods Appl. Mech. Engrg., 197 (2008), 806-820.  doi: 10.1016/j.cma.2007.09.009.  Google Scholar

[10]

M.-Y. Kim and M. F. Wheeler, A multiscale discontinuous Galerkin methods for convection-diffusion-reaction problems, Comput. Math. Appl., 68 (2014), 2251-2261.  doi: 10.1016/j.camwa.2014.08.007.  Google Scholar

[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.  Google Scholar

[12]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), 856-869.  doi: 10.1137/0907058.  Google Scholar

[13]

Ch. Schwab, p- and hp-finite element methods, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

show all references

References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[2]

P. Bochev, T. J. R. Hughes and G. Scovazzi, A multiscale discontinuous Galerkin method, Large-Scale Scientific Computing, Lecture Notes in Comput. Sci, 3743 (2006), 84-93.  doi: 10.1007/11666806_8.  Google Scholar

[3]

A. Buffa, T. J. R. Hughes and G. Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44 (2006), 1420-1440.  doi: 10.1137/050640382.  Google Scholar

[4]

E. T. Chung and W. T. Leung, A sub-grid structure enhanced discontinuous galerkin method for multiscale diffusion and convection-diffusion problems, Commun. Comput. Phys., 14 (2013), 370-392.  doi: 10.4208/cicp.071211.070912a.  Google Scholar

[5]

P. Houston, C. Schwab and E. Süli, Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.  doi: 10.1137/S0036142900374111.  Google Scholar

[6]

T. J. R. Hughes, G. Scovazzi, P. B. Bochev and A. Buffa, A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2761-2787.  doi: 10.1016/j.cma.2005.06.006.  Google Scholar

[7]

S. J. Jeong, A multiscale discontinuous Galerkin method for convection-diffusion-reaction problems: A numberical study, PhD Thesis. Google Scholar

[8]

S.-J. Jeong, M.-Y. Kim and T. Selenge, hp-discontinuous Galerkin methods for the Lotka-McKendrick equation$:$ A numerical study, Commun. Korean Math. Soc., 22 (2007), 623-640.  doi: 10.4134/CKMS.2007.22.4.623.  Google Scholar

[9]

D. Kim and E.-J. Park, A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems, Comput. Methods Appl. Mech. Engrg., 197 (2008), 806-820.  doi: 10.1016/j.cma.2007.09.009.  Google Scholar

[10]

M.-Y. Kim and M. F. Wheeler, A multiscale discontinuous Galerkin methods for convection-diffusion-reaction problems, Comput. Math. Appl., 68 (2014), 2251-2261.  doi: 10.1016/j.camwa.2014.08.007.  Google Scholar

[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.  Google Scholar

[12]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), 856-869.  doi: 10.1137/0907058.  Google Scholar

[13]

Ch. Schwab, p- and hp-finite element methods, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

Figure 1.  Schematic illustration of the basis functions in the local problem. The left hand side figure is a 16-node bicubic quadrilateral element. Its boundary nodes are identified on the one of the right hand side. The corresponding basis functions satisfy $ \overline{\phi}_j = \phi_j $, $ j = 1,2,\ldots, 12 $. The internal degrees-of-freedom, corresponding to $ \phi_{13} $, $ \phi_{14} $, $ \phi_{15} $, $ \phi_{16} $ are eliminated by the solution of the local problem. Only the unique, shared, boundary degrees-of-freedom are retained in the global problem (see [6])
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Exact solution when $ k = 10^{-4} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  DG solution with $ h = 1/32 $ and d.o.f $ = 6,144 $. Oscillation occurs due to convection dominant to $ k = 10^{-4} $ (diffusion coefficient)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  $ k = 10^{-4} $, $ h = 1/128 $, and $ P_1 $ element
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  $ k = 10^{-4} $, $ h = 1/32 $, and $ P_4 $ element
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  $ k = 10^{-6} $, $ h = 1/256 $, and $ P_1 $ element. DG solution with d.o.f $ = 393,216 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  $ k = 10^{-6} $, $ h = 1/256 $, and $ P_4 $ element
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.   
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  MDG solution with $ k = 10^{-6} $ and $ P_1 $ element
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Domain with uniform mesh $ h = 1/64 $. In grid and shadow parts, $ P_1 $ and $ P_4 $ elements are used, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  LHS: $ P_1 $, $ P_4 $ elements and triangle numbers. RHS: Matrix structure of DG corresponding to the triangles numbered on LHS
Figure Options

Download full-size image

Download as PowerPoint slide

Figure .  Vertexes of C0 constraints on the mismatched inner boundary
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  MDG solution $ \overline{u}^{MDG}_h $ with $ k = 10^{-6} $ using $ P_1 $ and $ P_4 $ elements
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Cases of $ P_1 $ and $ P_4 $ elements with $ k = 10^{-4} $ using GMRES with tol = $ 10^{-6} $
$ h $ Degree of freedom $ L^2 $ error Convergence order Degree order
1/32 6,144 8.66085e-002 $ \cdot $ 1
1/64 24,576 1.28362e-002 2.7543 1
1/128 98,304 1.77764e-003 2.8522 1
1/32 30,720 3.22321e-003 $ \cdot $ 4
(a) Using DG method
$h$ Degree of freedom $L^2$ error Convergence order Degree order
1/32 1,089 8.30570e-002 $\cdot$ 1
1/64 4,225 1.19648e-002 2.7953 1
1/128 16,641 1.87018e-003 2.6775 1
1/32 10,497 3.21210e-003 $\cdot$ 4
1/64 41,473 5.98660e-004 $\cdot$ 4
(b) Using MDG method
Table Options

Download as excel

$ h $ Degree of freedom $ L^2 $ error Convergence order Degree order
1/32 6,144 8.66085e-002 $ \cdot $ 1
1/64 24,576 1.28362e-002 2.7543 1
1/128 98,304 1.77764e-003 2.8522 1
1/32 30,720 3.22321e-003 $ \cdot $ 4
(a) Using DG method
$h$ Degree of freedom $L^2$ error Convergence order Degree order
1/32 1,089 8.30570e-002 $\cdot$ 1
1/64 4,225 1.19648e-002 2.7953 1
1/128 16,641 1.87018e-003 2.6775 1
1/32 10,497 3.21210e-003 $\cdot$ 4
1/64 41,473 5.98660e-004 $\cdot$ 4
(b) Using MDG method
Table 2.  DG approximation with $ P_1 $ and $ P_4 $ elements, $ k = 10^{-6} $, tol = $ 10^{-8} $ using GMRES with/without ILU (see [11,12])
$ h $ Total element num Degree of freedom $ L^2 $ error Convergence order Degree
1/64 8,192 24,576 6.6217e–001 $ \cdot $ 1
1/128 32,768 98,304 3.1138e–001 1.0885 1
1/256 131,072 393,216 9.9196e–002 1.6503 1
1/512 524,288 1,572,864 2.1732e–002 2.1911 1
1/1024 2,097,152 6,291,456 $ \cdot $ $ \cdot $ 1
1/256 131,072 1,966,080 6.4686e–003 $ \cdot $ 4
(a) DG solution
h Elapsed time GMRES iter(O/I) Elapsed time PGMRES Iter(O/I) Degree
1/64 8.2306e+001 1/208 7.2131e+000 1/4 1
1/128 1.4060e+003 3/201 6.1875e+001 1/4 1
1/256 2.5076e+004 10/220 1.4706e+003 1/4 1
1/512 $\cdot$ $\cdot$ 2.1915e+004 1/4 1
1/1024 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 1
1/256 4.9732e+005 10/256 3.1762e+004 1/10 4
(b) Comparison of GMRES with/without ILU for the DG in (a)
Table Options

Download as excel

$ h $ Total element num Degree of freedom $ L^2 $ error Convergence order Degree
1/64 8,192 24,576 6.6217e–001 $ \cdot $ 1
1/128 32,768 98,304 3.1138e–001 1.0885 1
1/256 131,072 393,216 9.9196e–002 1.6503 1
1/512 524,288 1,572,864 2.1732e–002 2.1911 1
1/1024 2,097,152 6,291,456 $ \cdot $ $ \cdot $ 1
1/256 131,072 1,966,080 6.4686e–003 $ \cdot $ 4
(a) DG solution
h Elapsed time GMRES iter(O/I) Elapsed time PGMRES Iter(O/I) Degree
1/64 8.2306e+001 1/208 7.2131e+000 1/4 1
1/128 1.4060e+003 3/201 6.1875e+001 1/4 1
1/256 2.5076e+004 10/220 1.4706e+003 1/4 1
1/512 $\cdot$ $\cdot$ 2.1915e+004 1/4 1
1/1024 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 1
1/256 4.9732e+005 10/256 3.1762e+004 1/10 4
(b) Comparison of GMRES with/without ILU for the DG in (a)
Table 3.  MDG approximation with $ P_1 $ and $ P_4 $ elements, $ k = 10^{-6} $, tol = $ 10^{-8} $ using GMRES with/without ILU
$ h $ Total element num Degree of freedom $ L^2 $ error convergence order Degree
1/64 8,192 4,225 6.5406e–001 $ \cdot $ 1
1/128 32,768 16,641 3.0008e–001 1.1241 1
1/256 131,072 66,049 9.3895e–002 1.6762 1
1/512 524,288 263,169 2.1614e–002 2.1191 1
1/1024 2,097,152 1,050,625 6.0343e–003 1.8417 1
1/256 131,072 657,409 6.3296e–003 $ \cdot $ 4
(a) MDG solution
h Elapsed time GMRES iter(O/I) Elapsed time PGMRES iter(O/I) Degree
1/64 1.2899e+001 1/160 5.8968e+000 1/9 1
1/128 1.1022e+002 1/246 4.4625e+001 1/12 1
1/256 2.4963e+003 3/189 2.1713e+002 1/14 1
1/512 3.0008e+004 5/125 2.7705e+003 1/18 1
1/1024 $\cdot$ $\cdot$ 2.5316e+005 1/20 1
1/256 6.5098e+004 10/256 4.8902e+003 1/25 4
(b) Comparison of GMRES with/without ILU for the MDG in (a)
Table Options

Download as excel

$ h $ Total element num Degree of freedom $ L^2 $ error convergence order Degree
1/64 8,192 4,225 6.5406e–001 $ \cdot $ 1
1/128 32,768 16,641 3.0008e–001 1.1241 1
1/256 131,072 66,049 9.3895e–002 1.6762 1
1/512 524,288 263,169 2.1614e–002 2.1191 1
1/1024 2,097,152 1,050,625 6.0343e–003 1.8417 1
1/256 131,072 657,409 6.3296e–003 $ \cdot $ 4
(a) MDG solution
h Elapsed time GMRES iter(O/I) Elapsed time PGMRES iter(O/I) Degree
1/64 1.2899e+001 1/160 5.8968e+000 1/9 1
1/128 1.1022e+002 1/246 4.4625e+001 1/12 1
1/256 2.4963e+003 3/189 2.1713e+002 1/14 1
1/512 3.0008e+004 5/125 2.7705e+003 1/18 1
1/1024 $\cdot$ $\cdot$ 2.5316e+005 1/20 1
1/256 6.5098e+004 10/256 4.8902e+003 1/25 4
(b) Comparison of GMRES with/without ILU for the MDG in (a)
Table 4.  MDG solution $ \overline{u}^{MDG}_h $ when $ k = 10^{-6} $, tol = $ 10^{-8} $, using ILU GMRES
$ h $ Ele. num. Basis num. Elapsed time $ L^2 $ error Conv. Iter.(O/I) Deg.
1/64 8,192 4,225 1.0632e+001 2.0077e–001 $ \cdot $ 1/9 1
1/128 32,768 16,641 7.3629e+001 8.1290e–002 1.3044 1/11 1
1/256 131,072 66,049 2.2198e+002 2.8772e–002 1.4984 1/13 1
1/512 524,288 263,169 2.3875e+003 6.5147e–003 2.1429 1/16 1
Using mixed polynomials (P1 and P2 elements)
Table Options

Download as excel

$ h $ Ele. num. Basis num. Elapsed time $ L^2 $ error Conv. Iter.(O/I) Deg.
1/64 8,192 4,225 1.0632e+001 2.0077e–001 $ \cdot $ 1/9 1
1/128 32,768 16,641 7.3629e+001 8.1290e–002 1.3044 1/11 1
1/256 131,072 66,049 2.2198e+002 2.8772e–002 1.4984 1/13 1
1/512 524,288 263,169 2.3875e+003 6.5147e–003 2.1429 1/16 1
Using mixed polynomials (P1 and P2 elements)
[1]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
[2]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265
[3]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169
[4]

Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029
[5]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003
[6]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[7]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[8]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[9]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[10]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[11]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103
[12]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126
[13]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[14]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[15]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031
[16]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[17]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[18]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049
[19]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053
[20]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054

 Impact Factor: 0.263

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences