doi: 10.3934/era.2020101

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

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.

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. ArnoldF. BrezziB. 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

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P. BochevT. 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. BuffaT. 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

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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

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P. HoustonC. 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. HughesG. ScovazziP. 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

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

[8]

S.-J. JeongM.-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. ArnoldF. BrezziB. 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. BochevT. 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. BuffaT. 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. HoustonC. 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. HughesG. ScovazziP. 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. JeongM.-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 2.  Exact solution when $ k = 10^{-4} $
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 4.  $ k = 10^{-4} $, $ h = 1/128 $, and $ P_1 $ element
Figure 5.  $ k = 10^{-4} $, $ h = 1/32 $, and $ P_4 $ element
Figure 6.  $ k = 10^{-6} $, $ h = 1/256 $, and $ P_1 $ element. DG solution with d.o.f $ = 393,216 $
Figure 7.  $ k = 10^{-6} $, $ h = 1/256 $, and $ P_4 $ element
Figure 9.  MDG solution with $ k = 10^{-6} $ and $ P_1 $ element
Figure 10.  Domain with uniform mesh $ h = 1/64 $. In grid and shadow parts, $ P_1 $ and $ P_4 $ elements are used, respectively
Figure 11.  LHS: $ P_1 $, $ P_4 $ elements and triangle numbers. RHS: Matrix structure of DG corresponding to the triangles numbered on LHS
Figure .  Vertexes of C0 constraints on the mismatched inner boundary
Figure 12.  MDG solution $ \overline{u}^{MDG}_h $ with $ k = 10^{-6} $ using $ P_1 $ and $ P_4 $ elements
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
$ 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)
$ 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)
$ 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)
$ 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)
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