Finite Element Method for Fractional Parabolic Integro-Differential Equations with Smooth and Nonsmooth Initial Data

We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in \({\mathbb {R}}^d (d=1,2,3)\). Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when \(u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )\). Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function \(u_0\) is only in \( L^2(\varOmega )\). The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.