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

Anouar El Harrak; Hatim Tayeq; Amal Bergam

doi:10.3934/dcdss.2021062

Article Contents

2021, Volume 14, Issue 7: 2183-2197. Doi: 10.3934/dcdss.2021062

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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
^* Corresponding author: anouarelharrak1@gmail.com

Received: November 30, 2019

Revised: May 31, 2020

Early access: May 2021

Published: July 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65M08, 35K60, 92D25; Secondary: 65N08, 65M15, 35K57.

Citation:

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Figure 1. 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 $

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Figure 2. Self-adapted meshes of three levels at instant $ t = 1 $

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Figure 3. Self-adapted meshes of three levels at instant $ t = 10 $

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Table 1. 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 $

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