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 decreasing including the case of non-Lipschitz or even non-H\"older 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.

翻译：本文针对具有潜在非Lipschitz非线性项的半线性椭圆方程，建立了有限元离散化的数值分析理论。非线性项本质上被假定为连续且单调递减，包含非Lipschitz甚至非Hölder连续的情形。对于采用线性有限元的直接离散化方法（无需任何正则化处理），我们推导了多种范数意义下的误差估计，并通过数值算例验证了理论结果。