Global Superconvergence and A Posteriori Error Estimators of the Finite Element Method for a Quasi-linear Elliptic Boundary Value Problem of Nonmonotone Type

In this paper we are concerned with finite element approximations to a nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions. This kind of problems arises for example from modeling a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree k \geq 1 in each variable, by means of an interpolation postprocessing technique, we obtain the global superconvergence of O(hk + 1) in the H1-norm and O(hk + 2) in the L2-norm provided the weak solution is sufficiently smooth. As by-products, the global superconvergence results can be used to generate efficient a posteriori error estimators. Representative numerical examples are also given to illustrate our theoretical analysis.

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