Numerical Approximation of Semilinear Subdiffusion Equations with Nonsmooth Initial Data

We consider the numerical approximation of a semilinear fractional order evolution equation involving a Caputo derivative in time of order $\alpha\in (0, 1)$. Assuming a Lipschitz continuous nonlinear source term and an initial data $u_0\in \dot H^\nu(\Omega)$, $\nu\in [0,2]$, we discuss existence and stability and provide regularity estimates for the solution of the problem. For a spatial discretization via piecewise linear finite elements, we establish optimal $L^2(\Omega)$-error estimates for cases with smooth and nonsmooth initial data, extending thereby known results derived for the classical semilinear parabolic problem. We further investigate fully implicit and linearized time-stepping schemes based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal $L^2(\Omega)$-error estimates for both numerical schemes. Numerical examples in one- and two-dimensional domains are presented to illustrate the theoretical results.