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.