A Second Order Non-Uniform IMEX-Alikhanov-FEM for Time-Fractional PDEs and PIDEs with Time-Dependent Coefficients

The stability and error analysis of a non-uniform implicit-explicit Alikhanov finite element method (IMEX-Alikhanov-FEM) are investigated for a class of time-fractional linear partial differential/integro-differential equations. These equations involve a non-self-adjoint elliptic operator with variable coefficients in both space and time. A second-order error estimate in the L 2-norm is derived, up to a logarithmic factor, for the problem with initial data in H 1 0 (Ω) ∩ H 2 (Ω). Furthermore, in the case of a self-adjoint elliptic operator , a superconvergence result is obtained in the H 1-norm, leading to an error estimate in the L ∞-norm for two-dimensional problems. All the estimates are α-robust, i.e., constants do not blow up as α → 1 −. A set of numerical experiments is conducted to confirm our theoretical results. AMS subject classifications. 65N35, 65N22