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Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion
July  2021, 14(7): 2183-2197. doi: 10.3934/dcdss.2021062

## A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics

 MAE2D laboratory, University Abdelmalek Essaadi, Polydisciplinary Faculty of Larache, Road of Rabat, Larache, Morocco

* Corresponding author: anouarelharrak1@gmail.com

Received  December 2019 Revised  June 2020 Published  July 2021 Early access  May 2021

This work gives a posteriori error estimates for a finite volume implicit scheme, applied to a two-time nonlinear reaction-diffusion problem in population dynamics, whose evolution processes occur at two different time scales, represented by a parameter $\varepsilon>0$ small enough. This work consists of building error indicators concerning time and space approximations and using them as a tool of adaptive mesh refinement in order to find approximate solutions to such models, in population dynamics, that are often hard to be handled analytically and also to be approximated numerically using the classical approach.

An application of the theoretical results is provided to emphasize the efficiency of our approach compared to the classical one for a spatial inter-specific model with constant diffusivity and population growth given by a logistic law in population dynamics.

Citation: Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
##### References:
 [1] B. Amaziane, A. Bergam, M. El Ossmani and Z. Mghazli, A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, J. Numer. Meth. Fluids, 59 (2009), 259-284.  doi: 10.1002/fld.1456. [2] P. Auger and R. Bravo, Methods of aggregation of variables in population dynamics, Comptes Rendus de l'Académie des Sciences - Series Ⅲ - Sciences de la Vie, 323 (2000), 665-674.  doi: 10.1016/S0764-4469(00)00182-7. [3] P. Auger, J. C. Poggiale and E. Sánchez, A review on spatial aggregation methods involving several time scales, Ecological Complexity, 10 (2012), 12-25.  doi: 10.1016/j.ecocom.2011.09.001. [4] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp, 74 (2005), 1117-1138.  doi: 10.1090/S0025-5718-04-01697-7. [5] A. Bergam, A. Chakib, A. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Applied Mathematics and Computation, 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069. [6] A. El Harrak and A. Bergam, Preserving Finite-Volume Schemes for Two-Time Reaction-Diffusion Model, Applied Mathematics & Information Sciences, 14 (2020), 41-50.  doi: 10.18576/amis/140105. [7] R. Eymard, T. Gallouöt and R. Herbin, Finite volume methods, Hand-book of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds, North-Holland, Ⅶ (2000), 713-1020.  doi: 10.1086/phos.67.4.188705. [8] M. Fortin, Estimation a posteriori et adaptation de maillages, Revue Européenne des Eléments Finis, 9 (2000). [9] W. Gurney and R. M. Nisbet, Ecological Dynamics,, Oxford University Press, 1998. [10] Z. Mghazli, R. Verfürth and A. Bergam, Estimateurs a posteriori d'un schéma de volumes finis pour un probléme non linéaire, Numerische Mathematik, 95 (2003), 599-624.  doi: 10.1007/s00211-003-0460-2. [11] J. D. Murray, Mathematical Biology: Ⅰ. an Introduction (Interdisciplinary Applied Mathematics), , Springer-Verlag, New York, 2002. [12] E. Sánchez, P. Auger and J. C. Poggiale, Two-time scales in spatially structured models of population dynamics: A semigroup approach, Journal of Mathematical Analysis and Applications, 375 (2011), 149-165.  doi: 10.1016/j.jmaa.2010.08.014. [13] D. Tilman and P. Kareiva, Spatial ecology: The role of space in population dynamics and interspecific interactions (MPB-30), Princeton University Press, 30 (2018). [14] R. Verfürth, A posteriori error estimates for non-stationary non-linear convection-diffusion equations, Calcolo, 55 (2018), Paper No. 20, 18 pp. doi: 10.1007/s10092-018-0263-6.

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##### References:
 [1] B. Amaziane, A. Bergam, M. El Ossmani and Z. Mghazli, A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, J. Numer. Meth. Fluids, 59 (2009), 259-284.  doi: 10.1002/fld.1456. [2] P. Auger and R. Bravo, Methods of aggregation of variables in population dynamics, Comptes Rendus de l'Académie des Sciences - Series Ⅲ - Sciences de la Vie, 323 (2000), 665-674.  doi: 10.1016/S0764-4469(00)00182-7. [3] P. Auger, J. C. Poggiale and E. Sánchez, A review on spatial aggregation methods involving several time scales, Ecological Complexity, 10 (2012), 12-25.  doi: 10.1016/j.ecocom.2011.09.001. [4] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp, 74 (2005), 1117-1138.  doi: 10.1090/S0025-5718-04-01697-7. [5] A. Bergam, A. Chakib, A. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Applied Mathematics and Computation, 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069. [6] A. El Harrak and A. Bergam, Preserving Finite-Volume Schemes for Two-Time Reaction-Diffusion Model, Applied Mathematics & Information Sciences, 14 (2020), 41-50.  doi: 10.18576/amis/140105. [7] R. Eymard, T. Gallouöt and R. Herbin, Finite volume methods, Hand-book of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds, North-Holland, Ⅶ (2000), 713-1020.  doi: 10.1086/phos.67.4.188705. [8] M. Fortin, Estimation a posteriori et adaptation de maillages, Revue Européenne des Eléments Finis, 9 (2000). [9] W. Gurney and R. M. Nisbet, Ecological Dynamics,, Oxford University Press, 1998. [10] Z. Mghazli, R. Verfürth and A. Bergam, Estimateurs a posteriori d'un schéma de volumes finis pour un probléme non linéaire, Numerische Mathematik, 95 (2003), 599-624.  doi: 10.1007/s00211-003-0460-2. [11] J. D. Murray, Mathematical Biology: Ⅰ. an Introduction (Interdisciplinary Applied Mathematics), , Springer-Verlag, New York, 2002. [12] E. Sánchez, P. Auger and J. C. Poggiale, Two-time scales in spatially structured models of population dynamics: A semigroup approach, Journal of Mathematical Analysis and Applications, 375 (2011), 149-165.  doi: 10.1016/j.jmaa.2010.08.014. [13] D. Tilman and P. Kareiva, Spatial ecology: The role of space in population dynamics and interspecific interactions (MPB-30), Princeton University Press, 30 (2018). [14] R. Verfürth, A posteriori error estimates for non-stationary non-linear convection-diffusion equations, Calcolo, 55 (2018), Paper No. 20, 18 pp. doi: 10.1007/s10092-018-0263-6.
Plot of approximate solution of Problem (12) using uniform mesh, ($A$) and ($C$), and adaptive mesh, ($B$) and ($D$), at two instants $t = 1$ and $t = 10$
Self-adapted meshes of three levels at instant $t = 1$
Self-adapted meshes of three levels at instant $t = 10$
Numerical tests for Problem (12)
 Instant Mesh Level Number of triangles CPU time Mean of $\eta_h^n$ t=1 Adaptive 1 $256$ $1.957s$ $3.1229e-2$ 2 $498$ $2.321s$ $1.2628e-2$ 3 $1071$ $5.492s$ $4.5722e-3$ Uniform 1 $6145$ $89.947s$ $3.3747e-3$ t=10 Adaptive 1 $207$ $9.861s$ $13053e-2$ 2 $504$ $16.087s$ $7.8965e-3$ 3 $726$ $23.411s$ $3.5019e-3$ Uniform 1 $6145$ $719.854s$ $1.1385e-3$ t=50 Adaptive 1 $356$ $44.871s$ $1.1824e-2$ 2 $603$ $76.644s$ $7.5797e-3$ 3 $879$ $120.120s$ $2.2937e-3$ Uniform 1 $6145$ $2686.813s$ $1.1418e-3$
 Instant Mesh Level Number of triangles CPU time Mean of $\eta_h^n$ t=1 Adaptive 1 $256$ $1.957s$ $3.1229e-2$ 2 $498$ $2.321s$ $1.2628e-2$ 3 $1071$ $5.492s$ $4.5722e-3$ Uniform 1 $6145$ $89.947s$ $3.3747e-3$ t=10 Adaptive 1 $207$ $9.861s$ $13053e-2$ 2 $504$ $16.087s$ $7.8965e-3$ 3 $726$ $23.411s$ $3.5019e-3$ Uniform 1 $6145$ $719.854s$ $1.1385e-3$ t=50 Adaptive 1 $356$ $44.871s$ $1.1824e-2$ 2 $603$ $76.644s$ $7.5797e-3$ 3 $879$ $120.120s$ $2.2937e-3$ Uniform 1 $6145$ $2686.813s$ $1.1418e-3$
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