error estimates for finite element discretization of semilinear elliptic equations with non-Lipschitz nonlinearities

In this paper we develop numerical analysis for finite element discretization of semilinear elliptic equations with potentially non-Lipschitz nonlinearites. The nonlinearity is essecially assumed to be continuous and monotonically non-decreasing including the case of non-Lipschitz or even non-Hölder continuous nonlinearities. For a direct discretization (without any regularization) with linear finite elements we derive error estimates with respect to various norms and illustrate these results with a numerical example.