February  2018, 11(1): 21-34. doi: 10.3934/dcdss.2018002

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

Received  September 2016 Revised  April 2017 Published  January 2018

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.

Citation: Hassan Belhadj, Mohamed Fihri, Samir Khallouq, Nabila Nagid. Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 21-34. doi: 10.3934/dcdss.2018002
References:
[1]

F. CaetanoM. J. GanderL. Halpern and J. Szeftel, Schwarz Waveform Relaxation Algorithms for Semilinear Reaction-Diffusion Equations, Networks And Heterog. Media, 5 (2010), 487-505. doi: 10.3934/nhm.2010.5.487. Google Scholar

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L. M. CarvalhoL. Giraud and P. Le Tallec, Algebraic two-level preconditioners for the Schur complement method, SIAM Journal on Scientific Computing, 22 (2000), 1987-2005. doi: 10.1137/S1064827598340809. Google Scholar

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T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations Oxford University, Clarendon Press, 1998. Google Scholar

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C. Chen and W. Liu, Two-grid finite volume element methods for semilinear parabolic problems, Applied Numerical Mathematics, 60 (2010), 10-18. doi: 10.1016/j.apnum.2009.08.004. Google Scholar

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M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap, SIAM Journal on Scientific Computing, 15 (1994), 604-620. doi: 10.1137/0915040. Google Scholar

[6]

S. DuminilH. Sadok and D. B. Szyld, Nonlinear Schwarz iterations with reduced rank extrapolation, Applied Numerical Mathematics, 94 (2015), 209-221. doi: 10.1016/j.apnum.2015.04.001. Google Scholar

[7]

R. I. Fernandes and G. Fairweather, An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems, Journal of Computational Physics, 231 (2012), 6248-6267. doi: 10.1016/j.jcp.2012.04.001. Google Scholar

[8]

C. Hay-Jahans, An R Companion to Linear Statistical Models CRC Press, 2011. doi: 10.1201/b11157. Google Scholar

[9]

F. N. Hochbruck and X. C. Cai, A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms, Computer methods in applied mechanics and engineering, 196 (2007), 1603-1611. doi: 10.1016/j.cma.2006.03.019. Google Scholar

[10]

S. Khallouq and H. Belhadj, Schur complement technique for advection-diffusion equation using matching structured finite volumes, Advances in Dynamical Systems and Applications, 8 (2013), 51-62. Google Scholar

[11]

R. KlajnM. FialkowskiI. T. BensemannA. BitnerC. J. CampbellK. BishopS. Smoukov and B. A. Grzybowski, Multicolour Micropatterning of Thin Films of Dry Gels, Nature materials, 3 (2004), 729-735. doi: 10.1038/nmat1231. Google Scholar

[12]

K. J. LeeW. D. McCormickJ. E. Pearson and H. L. Swinney, Experimental Observation of Self-Replicating Spots in a Reaction-Diffusion System, Nature, 369 (1994), 215-218. doi: 10.1038/369215a0. Google Scholar

[13]

J. W. Lottes and F. Fischer Paul, Hybrid multigrid/Schwarz algorithms for the spectral element method, Journal of Scientific Computing, 24 (2005), 45-78. doi: 10.1007/s10915-004-4787-3. Google Scholar

[14]

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations Oxford University Press, 1999. Google Scholar

[15]

A. Quarteroni and A. Valli, Theory and application of stecklov-poincaré operators for boundary value problems, In Applied and Industrial Mathematics. Springer Netherlands, 56 (1991), 179-203. Google Scholar

[16]

L. Roques, Equations de Réaction-Diffusion non-Linéaires et Modélisation en Ecologie Thesis of Univ. Pierre et Marie Curie, Paris 6,2004.Google Scholar

[17]

N. Shigesada and K. Kawasaki, Biological Invasions -Theory and Practice Oxford Series in Ecology and Evolution, Oxford University Press, 1997.Google Scholar

[18]

J. H. Stapleton, Linear Statistical Models John Wiley and Sons, 2009. Google Scholar

[19]

T. P. A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations Lecture Notes in Computational Science and Engineering, 61, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77209-5. Google Scholar

show all references

References:
[1]

F. CaetanoM. J. GanderL. Halpern and J. Szeftel, Schwarz Waveform Relaxation Algorithms for Semilinear Reaction-Diffusion Equations, Networks And Heterog. Media, 5 (2010), 487-505. doi: 10.3934/nhm.2010.5.487. Google Scholar

[2]

L. M. CarvalhoL. Giraud and P. Le Tallec, Algebraic two-level preconditioners for the Schur complement method, SIAM Journal on Scientific Computing, 22 (2000), 1987-2005. doi: 10.1137/S1064827598340809. Google Scholar

[3]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations Oxford University, Clarendon Press, 1998. Google Scholar

[4]

C. Chen and W. Liu, Two-grid finite volume element methods for semilinear parabolic problems, Applied Numerical Mathematics, 60 (2010), 10-18. doi: 10.1016/j.apnum.2009.08.004. Google Scholar

[5]

M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap, SIAM Journal on Scientific Computing, 15 (1994), 604-620. doi: 10.1137/0915040. Google Scholar

[6]

S. DuminilH. Sadok and D. B. Szyld, Nonlinear Schwarz iterations with reduced rank extrapolation, Applied Numerical Mathematics, 94 (2015), 209-221. doi: 10.1016/j.apnum.2015.04.001. Google Scholar

[7]

R. I. Fernandes and G. Fairweather, An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems, Journal of Computational Physics, 231 (2012), 6248-6267. doi: 10.1016/j.jcp.2012.04.001. Google Scholar

[8]

C. Hay-Jahans, An R Companion to Linear Statistical Models CRC Press, 2011. doi: 10.1201/b11157. Google Scholar

[9]

F. N. Hochbruck and X. C. Cai, A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms, Computer methods in applied mechanics and engineering, 196 (2007), 1603-1611. doi: 10.1016/j.cma.2006.03.019. Google Scholar

[10]

S. Khallouq and H. Belhadj, Schur complement technique for advection-diffusion equation using matching structured finite volumes, Advances in Dynamical Systems and Applications, 8 (2013), 51-62. Google Scholar

[11]

R. KlajnM. FialkowskiI. T. BensemannA. BitnerC. J. CampbellK. BishopS. Smoukov and B. A. Grzybowski, Multicolour Micropatterning of Thin Films of Dry Gels, Nature materials, 3 (2004), 729-735. doi: 10.1038/nmat1231. Google Scholar

[12]

K. J. LeeW. D. McCormickJ. E. Pearson and H. L. Swinney, Experimental Observation of Self-Replicating Spots in a Reaction-Diffusion System, Nature, 369 (1994), 215-218. doi: 10.1038/369215a0. Google Scholar

[13]

J. W. Lottes and F. Fischer Paul, Hybrid multigrid/Schwarz algorithms for the spectral element method, Journal of Scientific Computing, 24 (2005), 45-78. doi: 10.1007/s10915-004-4787-3. Google Scholar

[14]

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations Oxford University Press, 1999. Google Scholar

[15]

A. Quarteroni and A. Valli, Theory and application of stecklov-poincaré operators for boundary value problems, In Applied and Industrial Mathematics. Springer Netherlands, 56 (1991), 179-203. Google Scholar

[16]

L. Roques, Equations de Réaction-Diffusion non-Linéaires et Modélisation en Ecologie Thesis of Univ. Pierre et Marie Curie, Paris 6,2004.Google Scholar

[17]

N. Shigesada and K. Kawasaki, Biological Invasions -Theory and Practice Oxford Series in Ecology and Evolution, Oxford University Press, 1997.Google Scholar

[18]

J. H. Stapleton, Linear Statistical Models John Wiley and Sons, 2009. Google Scholar

[19]

T. P. A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations Lecture Notes in Computational Science and Engineering, 61, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77209-5. Google Scholar

Figure 1.  FV structured mesh of domain Ω
Figure 2.  Non-overlapping strip decomposition
Figure 3.  Domain decomposition and structured conforming mesh of domain Ω
Figure 4.  FE solution (left) and FV-1D (right), for h = 0:0138 and t = 0:5 in 3D
Figure 5.  FE solution (left) and FV-1D (right), for h = 0:0138 and t = 0:5 in 2D
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 $t=0.5$
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 $t=0.5$
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 $t=1$
Figure 9.  FV-1Dom solutions for different values of $h$, $y=0.0833$ (left) and $y=1.9167$ (right) at $t=0.5$
Figure 10.  FV-SC (2 subdomains) solutions for different values of $h$, $y=0.0833$ (left) and $y=1.9167$ (right) at $t=0.5$
Figure 11.  CPU time in seconds for FV-1Dom and FV-SC (h = 0:0138 and t = 1)
Figure 12.  Graphic representation of the adjustment to the power model
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
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
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)
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)
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)
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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