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Erschienen in:
2020 | OriginalPaper | Buchkapitel
A New Optimal \(L^{\infty }(H^1)\)–Error Estimate of a SUSHI Scheme for the Time Fractional Diffusion Equation
verfasst von : Abdallah Bradji
Erschienen in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples
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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].