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.

中文翻译：

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一类拟线性椭圆方程的理论和数值分析

本文的目的是给出在N维情况$ N \ geq 1 $中存在一个拟线性椭圆方程的非负弱解的结果，并给出一种新颖的数值方法来对其进行计算。在这项工作中，我们假定与解的导数有关的非线性是次二次的。设计用于计算所考虑方程的非负弱解的近似值的数值算法已将牛顿法与域分解和非线性的Yosida近似相结合。在牛顿法的每个步骤中，域分解都适用于非线性。给出了数值示例并进行了注释。