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:where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H 1 -norm by an indicator η 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 α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

“…The local lower bound can be proven in the similar way as that in . Thus, we omit it here.Theorem Assume that u and u h are the solutions of and , respectively.…”

“…In , Verfürth gave a general framework of a posteriori error estimates in H 1 ‐norm and L 2 ‐norm for nonlinear problems, respectively. When a ( u ) satisfies the strictly monotone assumption, a residual type of a posteriori error estimate in H 1 ‐norm for second‐order quasi‐linear elliptic equation was studied in . A posteriori error estimates for equations of prescribed mean curvature, that is, , were developed in .…”

“…The a posteriori error estimates of the finite element method for linear elliptic problems have been widely studied, see for instance and the references therein. The a posteriori error estimates of the finite element method for nonlinear elliptic problems have been considered in . In , Verfürth gave a general framework of a posteriori error estimates in H 1 ‐norm and L 2 ‐norm for nonlinear problems, respectively.…”