### A Posteriori Error Estimates of Residual Type for Second Order Quasi-Linear Elliptic PDEs

#### Abstract

We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator:

\begin{equation*}\label{pde}

\begin{array}{rcll}

-\nabla \cdot (\alpha(x,\nabla u)\nabla u)&=& f(x) ~~~~& \mbox{in}~\Omega\subset\mathbb{R}^2, \\

u&=& 0 &\mbox{on}~\partial\Omega,

\end{array}

\end{equation*}

where $\Omega$ is assumed to be a polygonal bounded domain in $\mathbb{R}^2$, $f \in L^2(\Omega)$, and $\alpha$ is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the $H^1$-norm by an indicator $\eta$ which is composed of $L^2$- norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the $\alpha$-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

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Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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