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Theoretical and numerical analysis of a class of quasilinear elliptic equations

  • * Corresponding author: Nahed Naceur

    * Corresponding author: Nahed Naceur 
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  • 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.

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


    \begin{equation} \\ \end{equation}
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  • Figure 1.  The computed super-solution $ {{w}_h} $

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

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

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

    Figure 5.  The final sub-domains decomposition

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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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