Numerical analysis
Notes on the convergence order of gradient schemes for time fractional differential equations
Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 439-448.

We apply the GDM (Gradient Discretization Method), developed recently, as discretization in space to time-fractional diffusion and diffusion-wave equations with a fractional derivative of Caputo type in any space dimension.

In the case of time-fractional diffusion equations, we establish an implicit scheme, and we prove an L(L2)-error estimate. A similar result in a discrete L(H01)–norm is also stated.

To construct the numerical scheme for the time-fractional diffusion-wave equation, we write the equation in the form of a system of two low-order equations. We state an a prior estimate result that helps us to derive error estimates in discrete semi-norms of L(H1) and H1(L2). The convergence is unconditional. Another gradient scheme is also suggested. We state its convergence results, which improve the convergence order proved recently for a SUSHI scheme.

These results hold then for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes.

On considère la méthode GD (Gradient Discretization), développée récemment, comme discrétisation dans l'espace pour les équations de diffusion et d'onde fractionnaires en temps avec une dérivée fractionnaire de type Caputo pour toute dimension d'espace. Le temps est discrétisé en intervalles de pas constant.

Dans le cas des équations de diffusion fractionnaires en temps, nous construisons un schéma implicite et nous prouvons une estimation d'erreur dans la norme L(L2). Un résultat similaire dans la norme L(H01)-norm a également été affirmé.

Pour construire le schéma numérique pour l'équation d'onde fractionnaire en temps, on écrit l'équation sous la forme d'un système de deux équations d'ordre plus bas. Nous construisons un schéma implicite, et nous donnons une estimation a priori qui nous permet de montrer des estimations d'erreur dans les semi-normes de L(H1) et H1(L2). La convergence est inconditionnelle. Un autre schéma a également été suggéré. Nous présentons ses résultats de convergence, qui améliorent l'ordre de convergence d'un schéma SUSHI prouvé récemment.

Les résultats obtenus sont alors valables pour tous les schémas particuliers de GDM : éléments finis, éléments finis mixtes, famille mimétique mixte hybride, quelques schémas de volumes finis multi-points et quelques schémas de Galerkin discontinus.

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DOI: 10.1016/j.crma.2018.02.006
Bradji, Abdallah 1

1 Department of Mathematics, University of Annaba, Algeria
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Bradji, Abdallah. Notes on the convergence order of gradient schemes for time fractional differential equations. Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 439-448. doi : 10.1016/j.crma.2018.02.006. http://www.numdam.org/articles/10.1016/j.crma.2018.02.006/

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