Optimal finite element approximations of monotone semilinear elliptic PDE with subcritical nonlinearities

We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori error estimate for a contractive Picard type iteration scheme on meshes that are locally refined towards possible corner singularities in polygonal domains. Our analysis involves, in particular, an elliptic regularity result in weighted Sobolev spaces and the use of the Trudinger inequality, which is instrumental in dealing with subcritically growing nonlinearities. A series of numerical experiments confirm the accuracy and efficiency of our method.