A New Optimal $L^{\infty }(H^1)$–Error Estimate of a SUSHI Scheme for the Time Fractional Diffusion Equation

Abstract

We consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional diffusion equation) in any space dimension. The time discretization is performed using a uniform mesh. We prove a new discrete $L^\infty (H^1)$–a priori estimate. Such a priori estimate is proved thanks to the use of the new tool of the discrete Laplace operator developed recently in [7]. Thanks to this a priori estimate, we prove a new optimal convergence order in the discrete $L^\infty (H^1)$–norm. These results improve the ones of [1, 4] which dealt respectively with finite volume and GDM (Gradient Discretization Method) for the TFDE. In [4], we only proved a priori estimate and error estimate in the discrete $L^\infty (L^2)$–norm whereas in [1] we proved a priori estimate and error estimate in the discrete $L^2(H^1)$–norm. The a priori estimate as well as the error estimate presented here were stated without proof for the first time in [3, Remark 1, p. 443] in the context of the general framework of GDM and [2, Remark 1, p. 205] in the context of finite volume methods. They also were mentioned, as future works, in [1, Remark 4.1].