Numerical Approximation of Semi-linear Sub-diffusion Problems with Local Lipschitz Conditions

We consider the numerical approximation of a semi-linear fractional order evolution equation involving a Caputo derivative in time of order ?, 0 ? 1. Assuming a locally Lipschitz continuous nonlinear source term and an initial data u0, we discuss existence, stability, and provide regularity estimates for the solution of the problem. For a spatial discretization via piecewise linear finite elements, we shall establish optimal error estimates for cases with smooth and non-smooth initial data, extending thereby known results derived for the standard semi-linear parabolic problem. We shall further investigate fully implicit and linearized time-stepping schemes based on a convolution quadrature in time generated by the backward Euler method. Our main objective is to provide pointwise-in-time optimal L2(?)-error estimates for both numerical schemes. Numerical examples will be conducted in one- and two-dimensional domains to illustrate the theoretical results.