In this communication we give a result of existence and present a numerical analysis of weak solutions for the following quasi-linear elliptic problem in one and two dimensions: \begin{equation} \;\;\;\;\;\left\{ \begin{array}{l} -{{A} }u(x)+G(x,{{D}} u(x))=F(x,u(x))+f(x)\hbox{\ \ in }\Omega\,, \\ u(x)=0\hbox{\ \ on }\partial \Omega \end{array} \right. \label{eq:1} \end{equation} where $A$ is a second order derivatives operator in one dimension and the Laplace operator in two dimensions, $G,F$ are Caratheodory non negative functions. The function $f$ is given finite and non negative. The domain $\Omega \subset \mathbb{R}^{N},\,\,N=1,2$ is open and bounded. Such problems arise from biological, chemical and physical systems and various methods have been proposed for study the existence, uniqueness, qualitative properties and numerical simulation of solutions. In the one dimensional we consider the case where $f$ is a non negative measure in $(0,1)$ ($f\in M_{B}^{+}(0,1)$) and where the growth of $G$ with respect to ${{D}}u=u^{\prime }$ and $F$ with respect to $u$ are arbitrary . In that case we prove that if there exists a supersolution then there exists a solution in $ W_{loc}^{1,\infty }(0,1)\bigcap C_{0}[0,1]$. In the two dimensional case we assume $f \in L^1(\Omega)$ and that $G(x,s)$ is sub-quadratic with respect to $s$. Then the problem (\ref{eq:1}) has a solution in $W_{0}^{1,q}(\Omega )$ where $1\leq q<{{N} / {(N-1)}}$, $N \geq 2$, provided that the equation (\ref{eq:1}) has a super-solution in $W_{0}^{1,1}(\Omega ).$ In the last section of this communication we present the numerical iterative method to solve the problem (\ref {eq:1}). Formally the iterative method constructs a sequence of numerical solutions of the Yosida approximation of equation (\ref{eq:1}) with a first guess which is a supersolution of problem (\ref{eq:1}), see \cite{AR} \cite{AR2}. To compute the supersolution we introduce a domain decomposition method. The domain partition should be determined by the behavior of the non linearity $F(x,u)$.