Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach
This work provides a unified theoretical framework for error analysis of time-fractional PDEs, benefiting researchers in numerical analysis and computational science.
The paper derives optimal error estimates for semidiscrete Galerkin finite element approximations of time-fractional parabolic problems with Caputo derivatives, handling both smooth and nonsmooth initial data. The analysis is unified and applicable to various spatial discretizations, including conforming and nonconforming FEMs.
In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order $α$, $0< α<1$. We derive optimal error estimates for semidiscrete Galerkin FE type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FEMs, and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multi-term time-fractional model is discussed.