On a non-uniform \(\alpha \)-robust IMEX-L1 mixed FEM for time-fractional PIDEs

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 \(\alpha \rightarrow 1^{-}\), 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.