Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem

Hassan Belhadj; Mohamed Fihri; Samir Khallouq; Nabila Nagid

doi:10.3934/dcdss.2018002

Article Contents

2018, Volume 11, Issue 1: 21-34. Doi: 10.3934/dcdss.2018002

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Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem

1.
Université Abdelmalek Essaadi, Faculté des Sciences et Techniques de Tanger, Laboratoire de Mathématiques et Applications, BP 416,90000 Tanger, Maroc
2.
Université Moulay Ismail, Faculté des Sciences et Techniques d'Errachidia, Laboratoire de Mathématiques Informatique et Image, Equipe d'Analyse Mathématique et Numérique des EDPs et Applications, BP 509, Boutalamine, 52000 Errachidia, Maroc

* Corresponding author: N. Nagid
* Corresponding author: N. Nagid

Received: August 31, 2016

Revised: March 31, 2017

Published: February 2018

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Abstract

The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q≥2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.

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Mathematics Subject Classification: Primary: 65M08, 65M55; Secondary: 62J05.

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Figure 1. FV structured mesh of domain Ω

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Figure 2. Non-overlapping strip decomposition

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Figure 3. Domain decomposition and structured conforming mesh of domain Ω

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Figure 4. FE solution (left) and FV-1D (right), for h = 0:0138 and t = 0:5 in 3D

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Figure 5. FE solution (left) and FV-1D (right), for h = 0:0138 and t = 0:5 in 2D

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Figure 6. Point-wise errors between the solution by FV-1Dom and by FE method (with the basis function in P1 (left) and P2 (right)), for $h=0.0138$ and $t=0.5$

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Figure 7. Point-wise errors between the solution by FV-SC (2SD) and by FE method (with the basis function in P1 (left) and P2 (right)), for $h=0.0138$ and $t=0.5$

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Figure 8. $L^{\infty}(\Omega)$ error history between FE and FV-1Dom (left) (resp. FV-SC (right)), with the basis functions in P1 (FE-P1) and P2 (FE-P2), for $h=0.0138$ and $t=1$

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Figure 9. FV-1Dom solutions for different values of $h$, $y=0.0833$ (left) and $y=1.9167$ (right) at $t=0.5$

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Figure 10. FV-SC (2 subdomains) solutions for different values of $h$, $y=0.0833$ (left) and $y=1.9167$ (right) at $t=0.5$

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Figure 11. CPU time in seconds for FV-1Dom and FV-SC (h = 0:0138 and t = 1)

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Figure 12. Graphic representation of the adjustment to the power model

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Table 1. $L^{2}(\Omega)$ errors between FV and FE-P2 ($h_{FE}=0.0069$) for different values of $h$ and of the number of subdomains, at $t=0.5$

Number of Subdomains	0.1666	h 0.0138	0.0069
1 Domain	0.0243	1.7708E-4	9.8594E-5
2 Subdomains	0.0243	1.7221E-4	8.9848E-5
3 Subdomains	0.0243	1.7430E-4	6.9495E-5
4 Subdomains	0.0243	1.7600E-4	7.0203E-5
6 Subdomains	0.0243	1.7485E-4	7.1316E-5
8 Subdomains	-	1.7375E-4	6.9220E-5
9 Subdomains	-	1.7357E-4	7.0606E-5

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Table 2. CPU time calculation for FV-1Dom and FV-SC at t = 1

Number of Subdomains	0.1666	h 0.0138	0.0069
1 Domain	0.1139 (second)	0.6285 (hour)	1.9884 (hour)
2 Subdomains	0.4891 (second)	0.4084 (hour)	1.4347 (hour)
3 Subdomains	0.4602 (second)	0.3264 (hour)	1.3845 (hour)
4 Subdomains	0.4823 (second)	0.3198 (hour)	1.4279 (hour)
6 Subdomains	0.5081 (second)	0.3299 (hour)	1.6049 (hour)
8 Subdomains	-	0.3555 (hour)	1.1068 (hour)
9 Subdomains	-	0.3700 (hour)	1.1577 (hour)

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Table 3. Optimal number of subdomains for different values of h

Optimal Number of subdomains	h	CPU time
1 Domain	0.1666	0.2018 (second)
2 Subdomains	0.083333	1.2547 (second)
3 Subdomains	0.01388	0.1344 (hour)
4 Subdomains	0.012820	0.4736 (hour)
5 Subdomains	0.00952	1.0142 (hour)
6 Subdomains	0.00925	0.9637 (hour)
7 Subdomains	0.00793	0.9124 (hour)
8 Subdomains	0.00694	1.0741 (hour)
9 Subdomains	0.00347	1.1570 (hour)
10 Subdomains	0.002222	1.4655 (hour)
11 Subdomains	0.002164	1.5632 (hour)
12 Subdomains	0.0021367	1.5932 (hour)
13 Subdomains	0.002051	2.1130 (hour)
35 Subdomains	0.00029304	5.4254 (hour)

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