
-
Previous Article
On recent progress of single-realization recoveries of random Schrödinger systems
- ERA Home
- This Issue
-
Next Article
Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays
Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems
Department of Mathematics, Inha University, Incheon 22212, Korea |
We investigate the matrix structure of the discrete system of the multiscale discontinuous Galerkin method (MDG) for general second order partial differential equations [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[13] |
Ch. Schwab, p- and hp-finite element methods, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[13] |
Ch. Schwab, p- and hp-finite element methods, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998. |







Degree of freedom | Convergence order | Degree order | ||
1/32 | 6,144 | 8.66085e-002 | 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 | 4 | |
(a) Using DG method | ||||
Degree of freedom | Convergence order | Degree order | ||
1/32 | 1,089 | 8.30570e-002 | 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 | 4 | |
1/64 | 41,473 | 5.98660e-004 | 4 | |
(b) Using MDG method |
Degree of freedom | Convergence order | Degree order | ||
1/32 | 6,144 | 8.66085e-002 | 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 | 4 | |
(a) Using DG method | ||||
Degree of freedom | Convergence order | Degree order | ||
1/32 | 1,089 | 8.30570e-002 | 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 | 4 | |
1/64 | 41,473 | 5.98660e-004 | 4 | |
(b) Using MDG method |
Total element num | Degree of freedom | Convergence order | Degree | ||
1/64 | 8,192 | 24,576 | 6.6217e–001 | 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 | 1 | ||
1/256 | 131,072 | 1,966,080 | 6.4686e–003 | 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 | 2.1915e+004 | 1/4 | 1 | ||
1/1024 | 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) |
Total element num | Degree of freedom | Convergence order | Degree | ||
1/64 | 8,192 | 24,576 | 6.6217e–001 | 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 | 1 | ||
1/256 | 131,072 | 1,966,080 | 6.4686e–003 | 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 | 2.1915e+004 | 1/4 | 1 | ||
1/1024 | 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) |
Total element num | Degree of freedom | convergence order | Degree | ||
1/64 | 8,192 | 4,225 | 6.5406e–001 | 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 | 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 | 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) |
Total element num | Degree of freedom | convergence order | Degree | ||
1/64 | 8,192 | 4,225 | 6.5406e–001 | 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 | 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 | 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) |
Ele. num. | Basis num. | Elapsed time | Conv. | Iter.(O/I) | Deg. | ||
1/64 | 8,192 | 4,225 | 1.0632e+001 | 2.0077e–001 | 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) |
Ele. num. | Basis num. | Elapsed time | Conv. | Iter.(O/I) | Deg. | ||
1/64 | 8,192 | 4,225 | 1.0632e+001 | 2.0077e–001 | 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]