Optimal error analysis of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

TL;DR

This paper develops and analyzes a non-uniform IMEX-L1 finite element method for time-fractional PDEs and PIDEs, providing optimal error estimates and super convergence results, validated by numerical experiments.

Contribution

It introduces a novel approach managing the interaction of fractional derivative approximation and elliptic operators for optimal error estimates in fractional PDEs.

Findings

Global almost optimal error estimates in $L^2$ and $H^1$ norms.

Super convergence when the elliptic operator is self-adjoint.

Validation of theoretical results through numerical experiments.

Abstract

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, global almost optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in $H^1$-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an $L^\infty$ error estimate is obtained for 2D problems that too with the initial condition is in $ H_0^1(\Omega)\cap H^2(\Omega)$. All results proved in this paper are valid uniformly as $\alpha\longrightarrow 1^{-}$, where $\alpha$ is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.