An $L^\infty (H^1)$-Error Estimate for Gradient Schemes Applied to Time Fractional Diffusion Equations

Abstract

In this work, we extend the results of [3] to some second order time accurate GSs (Gradient Schemes) applied to a general TFDE (Time Fractional Diffusion Equation) with a space-dependent conductivity. The time fractional derivative is taken in the Caputo sense. The space discretization is performed using the general framework of GDM (Gradient Discretization Method) which encompasses several numerical methods. The approximation of the Caputo derivative is given by the known $L2-1_\sigma$-formula. We prove a new discrete $L^\infty (H^1)$-a priori estimate which, in turn, helps establishing a new $L^\infty (H^1)$-error estimate for the stated second order time accurate GSs. The GDM considered in this work is restricted to the cases of the numerical methods in which $\Vert \varPi _\mathcal {D}\cdot \Vert _{L^2(\varOmega )}$ is a norm, where $\varPi _\mathcal {D}$ is the reconstruction operator of the approximate functions in the space $L^2(\varOmega )$.

Supported by MCS team (LAGA Laboratory) of the “Université Sorbonne- Paris Nord”.