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.
Highlights
A posteriori error estimation began playing role in analyzing the accuracy of the numerical solution with a pioneering work of Babuska and Rheinboldt (Babuska, I., 1978)
We estimated the actual error in the H1-norm by an indicator η which is composed of L2norms of the element residual and the jump residual
A local estimator shows us how good the approximation performs, but sometimes acts as an indicator used to determine whether that local mesh should be refined
Summary
A posteriori error estimation began playing role in analyzing the accuracy of the numerical solution with a pioneering work of Babuska and Rheinboldt (Babuska, I., 1978). For the linear case we refer to the works of Morin, Nochetto, and Siebert (Morin, P., 2000), where the convergence for second order elliptic equations with piecewise constant coefficients and without lower order terms were investigated by using a technique originated by Dorfler (Dorfler, W., 1996) They introduced the notion of data oscillation meant to quantify information missed in projecting the residual with discrete functions which is a process associated with the finite element method. M., 2011) designed an adaptive finite element algorithm for solving quasi-linear elliptic problems based on a Kacanov iteration They estimated the problem residual instead of the actual error, which need a practical way to deal with the negative norm in the dual space H−1. We discuss about the adaptive algorithm in the last section
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