Finite Element Method for a Time-Fractional Bi-harmonic Problem

Fractional-order partial differential equations have received considerable attention in recent years, from both theoretical and computational points of view, due to their excellent capacity in capturing the dynamics of non-local physical processes. In this project, we propose and analyze a finite element method (FEM) for solving a time-fractional bi-harmonic problem involving a fractional Caputo derivative in time and a fourth-order elliptic operator in space. A detailed study of the regularity of the exact solution is included, which provides the essential information to carry out a rigorous error analysis. For a FEM spatial discretization via piecewise linear elements, we shall establish optimal error estimates for solutions with smooth and non-smooth initial data, extending thereby known results derived for the standard parabolic bi-harmonic problem. Completely discrete schemes are proposed using a convolution quadrature in time generated by the backward Euler method. We shall then investigate fully implicit and linearized semi implicit linearized time-stepping schemes. Our main objective is to provide pointwise-in-time optimal L2(?)-error estimates for both numerical schemes. To illustrate the theoretical results, several numerical tests will be provided in one- and two-dimensional domains.