On a Non-Uniform $\alpha$-Robust IMEX-L1 Mixed FEM for Time-Fractional PIDEs

TL;DR

This paper develops and analyzes a non-uniform IMEX-L1 mixed finite element method for solving time-fractional PIDEs, providing stability, optimal error estimates, and confirming results through numerical experiments.

Contribution

It introduces a novel non-uniform IMEX-L1 mixed FEM for time-fractional PIDEs with space-time dependent coefficients, including stability analysis and error estimates.

Findings

Optimal error estimates in $L^2$-norm for solution and flux.

Error estimate in $L^ Infinity$-norm for 2D problems.

Numerical experiments confirm theoretical results.

Abstract

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables. The focus of the study is to analyze stability results and to establish optimal error estimates, up to a logarithmic factor, for both the solution and the flux in $L^2$-norm when the initial data $u_0\in H_0^1(\Omega)\cap H^2(\Omega)$. Additionally, an error estimate in $L^\infty$-norm is derived for 2D problems. All the derived estimates and bounds in this article remain valid as $\alpha\to 1^{-}$, where $\alpha$ is the order of the Caputo fractional derivative. Finally, the results of several numerical experiments conducted at the end of this paper are confirming our theoretical findings.