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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  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 & Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
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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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