July  2021, 14(7): 2075-2099. doi: 10.3934/dcdss.2020260

Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique

1. 

Abdelmalek Essaadi University, Faculty of Sciences and Techniques of Tangier, Laboratory of Mathematics and Applications, Department of Mathematics, BP. 416, Tangier 90000, Morocco

2. 

Moulay Ismail University of Meknes Faculty of Sciences and Techniques of Errachidia Department of Mathematics MSISI Laboratory, AM2CSI Group BP. 509, Boutalamine Errachidia 57000, Morocco

3. 

Moulay Ismail University of Meknes, Faculty of Sciences of Meknes, Department of Mathematics and Informatics, BP. 11201 Zitoune Meknes, Morocco

* Corresponding author: Samir Khallouq

Received  March 2019 Revised  June 2019 Published  July 2021 Early access  September 2020

We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve $ 2D $ anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.

Citation: Hassan Belhadj, Samir Khallouq, Mohamed Rhoudaf. Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2075-2099. doi: 10.3934/dcdss.2020260
References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.

[2]

B. AndreianovF. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.  doi: 10.1002/num.20170.

[3]

H. BelhadjM. FihriS. Khallouq and N. Nagid, Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.  doi: 10.3934/dcdss.2018002.

[4]

P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.  doi: 10.1137/0723075.

[5]

F. BoyerF. Hubert and S. Krell, Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.  doi: 10.1093/imanum/drp001.

[6]

S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.  doi: 10.1007/s002110050446.

[7]

T. F. ChanE. Weinan and J. Sun, Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.  doi: 10.1016/0168-9274(91)90072-8.

[8]

Y. CoudièreJ.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.  doi: 10.1051/m2an:1999149.

[9]

K. Domelevo and P. Omnès, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.  doi: 10.1051/m2an:2005047.

[10]

R. EymardT. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.  doi: 10.1007/s002110050412.

[11]

R. EymardT. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.

[12]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020.

[13]

R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692.

[14]

F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.  doi: 10.1006/jcph.2000.6466.

[15]

L. Mansfield, On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.  doi: 10.1137/0727094.

[16] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999. 

show all references

References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.

[2]

B. AndreianovF. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.  doi: 10.1002/num.20170.

[3]

H. BelhadjM. FihriS. Khallouq and N. Nagid, Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.  doi: 10.3934/dcdss.2018002.

[4]

P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.  doi: 10.1137/0723075.

[5]

F. BoyerF. Hubert and S. Krell, Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.  doi: 10.1093/imanum/drp001.

[6]

S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.  doi: 10.1007/s002110050446.

[7]

T. F. ChanE. Weinan and J. Sun, Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.  doi: 10.1016/0168-9274(91)90072-8.

[8]

Y. CoudièreJ.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.  doi: 10.1051/m2an:1999149.

[9]

K. Domelevo and P. Omnès, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.  doi: 10.1051/m2an:2005047.

[10]

R. EymardT. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.  doi: 10.1007/s002110050412.

[11]

R. EymardT. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.

[12]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020.

[13]

R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692.

[14]

F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.  doi: 10.1006/jcph.2000.6466.

[15]

L. Mansfield, On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.  doi: 10.1137/0727094.

[16] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999. 
Figure 1.  DDFV mesh
Figure 2.  Diamond cell $ \mathcal{D}_{\sigma,\sigma^{*}} $
Figure 3.  A DDFV mesh $ \mathcal{T} $ of the whole domain $ \Omega $
Figure 4.  The compatible meshes $ \mathcal{T}_{1} $ and $ \mathcal{T}_{2} $, of the whole domain $ \Omega $, corresponding to the DDFV mesh $ \mathcal{T} $ of figure 3
Figure 5.  Two independent DDFV meshes $ \mathcal{T}_{1} $ and $ \mathcal{T}_{2} $ for both subdomains $ \Omega_{i} $
Figure 6.  The compatible meshes corresponding to two independent DDFV meshes $ \mathcal{T}_{1} $ and $ \mathcal{T}_{2} $ of figure 5
Figure 7.  Primal mesh $ Mesh_{1} $ (left) and $ Mesh_{2} $ (right)
Figure 8.  Analytic solution (left) and DDFV solution (right) for the test case $ 1 $ and $ h = 0.0118 $
Figure 9.  Analytic solution (left) and DDFV solution (right) for the test case $ 2 $ and $ h = 0.0207 $
Figure 10.  Primal mesh $ Mesh_{2,1} $ (left) and $ Mesh_{1,3} $ (right)
Figure 11.  CPU time to solve the test case $ 2 $ for different numbers of diamonds $ \mathcal{D} $
Table 1.  $ L^{\infty}(\Omega) $ and $ L^{2}(\Omega) $ errors by DDFV, for test case $ 1 $ and for different values of $ h $
Mesh Nbr of $ \mathcal{D} $ $ h $ $ e_{h,L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h,L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1} $ $ 488 $ $ 0.1184 $ $ 0.0037 $ - $ 5.3348 $E-$ 04 $ -
$ Mesh_{2} $ $ 1912 $ $ 0.0645 $ $ 8.5460 $E-$ 04 $ $ 2.4127 $ $ 1.3259 $E-$ 04 $ $ 2.2920 $
$ Mesh_{3} $ $ 7568 $ $ 0.0364 $ $ 2.5019 $E-$ 04 $ $ 2.1472 $ $ 3.3329 $E-$ 05 $ $ 2.4136 $
$ Mesh_{4} $ $ 30112 $ $ 0.0207 $ $ 8.2489 $E-$ 05 $ $ 1.9658 $ $ 8.3355 $E-$ 06 $ $ 2.4554 $
Average $ 2.1752 $ $ 2.3870 $
Mesh Nbr of $ \mathcal{D} $ $ h $ $ e_{h,L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h,L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1} $ $ 488 $ $ 0.1184 $ $ 0.0037 $ - $ 5.3348 $E-$ 04 $ -
$ Mesh_{2} $ $ 1912 $ $ 0.0645 $ $ 8.5460 $E-$ 04 $ $ 2.4127 $ $ 1.3259 $E-$ 04 $ $ 2.2920 $
$ Mesh_{3} $ $ 7568 $ $ 0.0364 $ $ 2.5019 $E-$ 04 $ $ 2.1472 $ $ 3.3329 $E-$ 05 $ $ 2.4136 $
$ Mesh_{4} $ $ 30112 $ $ 0.0207 $ $ 8.2489 $E-$ 05 $ $ 1.9658 $ $ 8.3355 $E-$ 06 $ $ 2.4554 $
Average $ 2.1752 $ $ 2.3870 $
Table 2.  $ L^{\infty}(\Omega) $ and $ L^{2}(\Omega) $ errors by DDFV, for test case $ 2 $ and for different values of $ h $
Mesh Nbr of $ \mathcal{D} $ $ h $ $ e_{h,L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h,L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1} $ $ 488 $ $ 0.1184 $ $ 0.2402 $ - $ 0.0452 $ -
$ Mesh_{2} $ $ 1912 $ $ 0.0645 $ $ 0.0423 $ $ 2.8592 $ $ 0.0106 $ $ 2.3876 $
$ Mesh_{3} $ $ 7568 $ $ 0.0364 $ $ 0.0132 $ $ 2.0356 $ $ 0.0028 $ $ 2.3269 $
$ Mesh_{4} $ $ 30112 $ $ 0.0207 $ $ 0.0048 $ $ 1.7922 $ $ 7.3969 $E-$ 04 $ $ 2.3584 $
Average $ 2.2290 $ $ 2.3576 $
Mesh Nbr of $ \mathcal{D} $ $ h $ $ e_{h,L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h,L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1} $ $ 488 $ $ 0.1184 $ $ 0.2402 $ - $ 0.0452 $ -
$ Mesh_{2} $ $ 1912 $ $ 0.0645 $ $ 0.0423 $ $ 2.8592 $ $ 0.0106 $ $ 2.3876 $
$ Mesh_{3} $ $ 7568 $ $ 0.0364 $ $ 0.0132 $ $ 2.0356 $ $ 0.0028 $ $ 2.3269 $
$ Mesh_{4} $ $ 30112 $ $ 0.0207 $ $ 0.0048 $ $ 1.7922 $ $ 7.3969 $E-$ 04 $ $ 2.3584 $
Average $ 2.2290 $ $ 2.3576 $
Table 3.  $ L^{\infty}(\Omega) $ and $ L^{2}(\Omega) $ errors by DDFV-SC, for the test case $ 1 $ and for different mesh $ Mesh_{i,i} $ with $ i = 1,\ldots,4 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1}\cup \Omega_{2} $ $ h $ $ e_{h_{1,2},L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h_{1,2},L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1,1} $ $ 546 $ $ 0.1191 $ $ 0.0020 $ $ - $ $ 5.4853 $E-$ 04 $ $ - $
$ Mesh_{2,2} $ $ 2124 $ $ 0.0626 $ $ 5.6512 $E-$ 04 $ $ 1.9650 $ $ 1.3729 $E-$ 04 $ $ 2.1535 $
$ Mesh_{3,3} $ $ 8376 $ $ 0.0354 $ $ 2.0197 $E-$ 04 $ $ 1.8050 $ $ 3.4357 $E-$ 05 $ $ 2.4301 $
$ Mesh_{4,4} $ $ 33264 $ $ 0.0200 $ $ 7.0126 $E-$ 05 $ $ 1.8527 $ $ 8.5752 $E-$ 06 $ $ 2.4308 $
Average $ 1.8742 $ $ 2.3381 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1}\cup \Omega_{2} $ $ h $ $ e_{h_{1,2},L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h_{1,2},L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1,1} $ $ 546 $ $ 0.1191 $ $ 0.0020 $ $ - $ $ 5.4853 $E-$ 04 $ $ - $
$ Mesh_{2,2} $ $ 2124 $ $ 0.0626 $ $ 5.6512 $E-$ 04 $ $ 1.9650 $ $ 1.3729 $E-$ 04 $ $ 2.1535 $
$ Mesh_{3,3} $ $ 8376 $ $ 0.0354 $ $ 2.0197 $E-$ 04 $ $ 1.8050 $ $ 3.4357 $E-$ 05 $ $ 2.4301 $
$ Mesh_{4,4} $ $ 33264 $ $ 0.0200 $ $ 7.0126 $E-$ 05 $ $ 1.8527 $ $ 8.5752 $E-$ 06 $ $ 2.4308 $
Average $ 1.8742 $ $ 2.3381 $
Table 4.  $ L^{\infty}(\Omega) $ and $ L^{2}(\Omega) $ errors by DDFV-SC, for the test case $ 2 $ and for different mesh $ Mesh_{i,i} $ with $ i = 1,\ldots,4 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1}\cup \Omega_{2} $ $ h $ $ e_{h_{1,2},L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h_{1,2},L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1,1} $ $ 546 $ $ 0.1191 $ $ 0.3694 $ $ - $ $ 0.0980 $ $ - $
$ Mesh_{2,2} $ $ 2124 $ $ 0.0626 $ $ 0.0733 $ $ 2.5145 $ $ 0.0152 $ $ 2.8975 $
$ Mesh_{3,3} $ $ 8376 $ $ 0.0354 $ $ 0.0191 $ $ 2.3592 $ $ 0.0036 $ $ 2.5267 $
$ Mesh_{4,4} $ $ 33264 $ $ 0.0200 $ $ 0.0053 $ $ 2.2452 $ $ 8.6492 $E-$ 04 $ $ 2.4976 $
Average $ 2.3730 $ $ 2.6406 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1}\cup \Omega_{2} $ $ h $ $ e_{h_{1,2},L^{\infty}} $ $ EOC_{L^{\infty}} $ $ e_{h_{1,2},L^{2}} $ $ EOC_{L^{2}} $
$ Mesh_{1,1} $ $ 546 $ $ 0.1191 $ $ 0.3694 $ $ - $ $ 0.0980 $ $ - $
$ Mesh_{2,2} $ $ 2124 $ $ 0.0626 $ $ 0.0733 $ $ 2.5145 $ $ 0.0152 $ $ 2.8975 $
$ Mesh_{3,3} $ $ 8376 $ $ 0.0354 $ $ 0.0191 $ $ 2.3592 $ $ 0.0036 $ $ 2.5267 $
$ Mesh_{4,4} $ $ 33264 $ $ 0.0200 $ $ 0.0053 $ $ 2.2452 $ $ 8.6492 $E-$ 04 $ $ 2.4976 $
Average $ 2.3730 $ $ 2.6406 $
Table 5.  $ L^{\infty}(\Omega) $ and $ L^{2}(\Omega) $ errors by DDFV-SC, for the test case $ 3 $ and for different mesh $ Mesh_{i,j} $ with $ i,j = 1,\ldots,4 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1} $ Nbr of $ \mathcal{D} $ in $ \Omega_{2} $ $ h_{1} $ $ h_{2} $ $ e_{h_{1,2},L^{\infty}} $ $ e_{h_{1,2},L^{2}} $
$ Mesh_{1,1} $ $ 273 $ $ 273 $ $ 0.1191 $ $ 0.1187 $ $ 0.0238 $ $ 0.0068 $
$ Mesh_{2,1} $ $ 1062 $ $ 283 $ $ 0.0626 $ $ 0.1187 $ $ 0.0195 $ $ 0.0053 $
$ Mesh_{2,2} $ $ 1062 $ $ 1062 $ $ 0.0626 $ $ 0.0621 $ $ 0.0129 $ $ 0.0035 $
$ Mesh_{3,2} $ $ 4188 $ $ 1082 $ $ 0.0354 $ $ 0.0621 $ $ 0.0103 $ $ 0.0028 $
$ Mesh_{3,3} $ $ 4188 $ $ 4188 $ $ 0.0354 $ $ 0.0347 $ $ 0.0066 $ $ 0.0018 $
$ Mesh_{4,3} $ $ 16632 $ $ 4228 $ $ 0.0200 $ $ 0.0347 $ $ 0.0053 $ $ 0.0014 $
$ Mesh_{4,4} $ $ 16632 $ $ 16632 $ $ 0.0200 $ $ 0.0191 $ $ 0.0033 $ $ 9.1142 $E-$ 04 $
Mesh Nbr of $ \mathcal{D} $ in $ \Omega_{1} $ Nbr of $ \mathcal{D} $ in $ \Omega_{2} $ $ h_{1} $ $ h_{2} $ $ e_{h_{1,2},L^{\infty}} $ $ e_{h_{1,2},L^{2}} $
$ Mesh_{1,1} $ $ 273 $ $ 273 $ $ 0.1191 $ $ 0.1187 $ $ 0.0238 $ $ 0.0068 $
$ Mesh_{2,1} $ $ 1062 $ $ 283 $ $ 0.0626 $ $ 0.1187 $ $ 0.0195 $ $ 0.0053 $
$ Mesh_{2,2} $ $ 1062 $ $ 1062 $ $ 0.0626 $ $ 0.0621 $ $ 0.0129 $ $ 0.0035 $
$ Mesh_{3,2} $ $ 4188 $ $ 1082 $ $ 0.0354 $ $ 0.0621 $ $ 0.0103 $ $ 0.0028 $
$ Mesh_{3,3} $ $ 4188 $ $ 4188 $ $ 0.0354 $ $ 0.0347 $ $ 0.0066 $ $ 0.0018 $
$ Mesh_{4,3} $ $ 16632 $ $ 4228 $ $ 0.0200 $ $ 0.0347 $ $ 0.0053 $ $ 0.0014 $
$ Mesh_{4,4} $ $ 16632 $ $ 16632 $ $ 0.0200 $ $ 0.0191 $ $ 0.0033 $ $ 9.1142 $E-$ 04 $
Table 6.  CPU time to solve the test case $ 1 $ for different numbers of diamonds $ \mathcal{D} $
Nbr of $ \mathcal{D} $ in $ \Omega $ CPU time for DDFV (in seconds) CPU time for DDFV-SC (in seconds)
$ 8376 $ $ 18.149494 $ $ 24.458821 $
$ 20860 $ $ 94.936105 $ $ 59.901873 $
$ 33264 $ $ 418.468599 $ $ 229.306277 $
$ 82920 $ $ 2899.437600 $ $ 1285.708950 $
$ 132576 $ $ 18312.136003 $ $ 3947.530428 $
$ 330960 $ $ 126004.456142 $ $ 7801.259200 $
Nbr of $ \mathcal{D} $ in $ \Omega $ CPU time for DDFV (in seconds) CPU time for DDFV-SC (in seconds)
$ 8376 $ $ 18.149494 $ $ 24.458821 $
$ 20860 $ $ 94.936105 $ $ 59.901873 $
$ 33264 $ $ 418.468599 $ $ 229.306277 $
$ 82920 $ $ 2899.437600 $ $ 1285.708950 $
$ 132576 $ $ 18312.136003 $ $ 3947.530428 $
$ 330960 $ $ 126004.456142 $ $ 7801.259200 $
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