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

Hassan Belhadj; Samir Khallouq; Mohamed Rhoudaf

doi:10.3934/dcdss.2020260

Article Contents

2021, Volume 14, Issue 7: 2075-2099. Doi: 10.3934/dcdss.2020260

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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
^* Corresponding author: Samir Khallouq

Received: March 2019

Revised: June 2019

Early access: September 2020

Published: July 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 76R50, 65N08, 65N55; Secondary: 65Y05.

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Figure 1. DDFV mesh

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Figure 2. Diamond cell $ \mathcal{D}_{\sigma,\sigma^{*}} $

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Figure 3. A DDFV mesh $ \mathcal{T} $ of the whole domain $ \Omega $

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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

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Figure 5. Two independent DDFV meshes $ \mathcal{T}_{1} $ and $ \mathcal{T}_{2} $ for both subdomains $ \Omega_{i} $

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Figure 6. The compatible meshes corresponding to two independent DDFV meshes $ \mathcal{T}_{1} $ and $ \mathcal{T}_{2} $ of figure 5

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Figure 7. Primal mesh $ Mesh_{1} $ (left) and $ Mesh_{2} $ (right)

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Figure 8. Analytic solution (left) and DDFV solution (right) for the test case $ 1 $ and $ h = 0.0118 $

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Figure 9. Analytic solution (left) and DDFV solution (right) for the test case $ 2 $ and $ h = 0.0207 $

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Figure 10. Primal mesh $ Mesh_{2,1} $ (left) and $ Mesh_{1,3} $ (right)

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Figure 11. CPU time to solve the test case $ 2 $ for different numbers of diamonds $ \mathcal{D} $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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