Error Estimates for FEMs for Time-Fractional Diffusion Problems

In recent years, the time-fractional sub-diffusion model has received much attention, due to its excellent capacity of capturing the dynamics of physical processes involving an anomalous transport mechanism. The limited solution regularity of the sub-diffusion model underlies the main technical difficulty in developing robust numerical schemes for approximating its solution. In this project, we propose a novel error analysis based on delicate energy arguments combined with a technique that exploits the property of the inverse of the associated elliptic problem. We shall present our method in a general setting so that it can extended to different spatial approximations, and can be easily adopted to several time-fractional problems. The method allows in particular the derivation of error estimates in various norms. We expect that our method can be extended to solve time depending mixed and Stokes problems. Numerical experiments will be conducted to confirm the theoretical results. Special attention will be given to the regularity of exact solution.