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

We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: egin{equation*}label{pde} egin{array}{rcll} - abla cdot (alpha(x, abla u) abla u)&=& f(x) ~~~~& mbox{in}~Omegasubsetmathbb{R}^2, u&=& 0 &mbox{on}~partialOmega, 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.