Theoretical and numerical analysis of a class of quasilinear elliptic equations

Nahed Naceur; Nour Eddine Alaa; Moez Khenissi; Jean R. Roche

doi:10.3934/dcdss.2020354

Article Contents

2021, Volume 14, Issue 2: 723-743. Doi: 10.3934/dcdss.2020354

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Theoretical and numerical analysis of a class of quasilinear elliptic equations

1.
Université de Lorraine, CNRS, IECL, F 54000 Nancy, France
2.
Laboratory LAMAI, Faculty of Science and Technology, University Cadi Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco
3.
LAMMDA, Université de Sousse, ESST Hammam Sousse, Rue Lamine El Abbessi, Hammam Sousse, 4011, Tunisia

^* Corresponding author: Nahed Naceur
^* Corresponding author: Nahed Naceur

Received: September 30, 2019

Revised: November 30, 2019

Early access: May 2020

Published: February 2021

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Abstract

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, $ N\geq 1 $, and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the non-negative weak solution of the considered equation has coupled the Newton method with domain decomposition and Yosida approximation of the nonlinearity. The domain decomposition is adapted to the nonlinearity at each step of the Newton method. Numerical examples are presented and commented on.

Keywords:

Mathematics Subject Classification: Primary: 35J62, 65M55, 65N30; Secondary: 49M15.

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Figure 1. The computed super-solution $ {{w}_h} $

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Figure 2. The numerical approximation of the solution of (30)

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Figure 3. Evolution of the error w.r.t. $ n $

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Figure 4. The numerical approximation of the super-solution of (31)

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Figure 5. The final sub-domains decomposition

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Figure 6. The computed solution at the 20th step of the algorithm implementing the Yosida's approximation of $ G $

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Table 1. The $L^2$-norm of error between $u$ and $u_{n, h}$

n	1	2	3	...	8	9	10
Error	$0.6$	$0.281$	0.1033	...	$1.2 \; 10^{-4}$	$3.05\; 10^{-5}$	$8.21 \; 10^{-6}$
#Newton iteration	5	7	9	...	7	6	6

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Table 2. The behavior of the algorithm computing the super-solution

Newton iteration	1	2	3	4
Norm of the Newton update	$ 4.21 $	$ 0.17 $	$ 0.026 $	$ 6.4\; 10^{-4} $
# Schwarz iteration	-	20	14	6
Norm of the Schwarz update	-	$ 8.6\; 10^{-4} $	$ 9.1\; 10^{-4} $	$ 8.8\; 10^{-4} $
#sub-domains	1	9	16	16

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Table 3. The behavior of the algorithm computing the solution of the problem (20)

n	1	2	...	6	7	...	19	20
Norm of the update	0.1009	0.1037	...	0.0232	0.018	...	0.0065	0.0062
# Newton iteration	7	8	...	8	9	...	9	8

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