A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

Abstract

We consider as discretization in space, the GDM (Gradient Discretization Method) developed recently in Droniou et al. (2018), to approximate multidimensional time fractional diffusion and diffusion-wave equations where the fractional order α of the time derivative is respectively satisfying 0<α<1 and 1<α<2. The time fractional derivative is given in the Caputo sense. The time discretization is performed using a uniform mesh. For the time fractional diffusion equation, we derive an implicit scheme. In addition to an L∞(L2)-a priori estimate, which can be derived from a reasoning in Bradji (2018), we present and prove a new L2(H01)-a priori estimate for the discrete problem. Under the assumption that the exact solution is sufficiently smooth, these a priori estimates allow to prove error estimates in discrete norms of L2(H01) and L∞(L2). The convergence order in these estimates is optimal in the sense of two points of view. The first one is that the order in space is the same one proved in Droniou et al. (2018) for elliptic equation, whereas the second point of view is that the particular case of this order when α=1 is the same one obtained in Droniou et al. (2018) for the standard heat equation. For the time fractional diffusion-wave equation, we derive two implicit schemes. A full convergence analysis is carried out for both schemes. In particular, we develop new a priori estimates which yield error estimates in several discrete norms for each scheme. The convergence is proved to be unconditional and optimal. The convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates. These results improve the ones of Bradji (2017) in which a conditional convergence is proved for a SUSHI (Scheme using Stabilization and Hybrid Interfaces) approximating a time-fractional diffusion-wave equation. For both cases of time fractional diffusion and diffusion-wave equations, we show the well-posedness in the sense that the discrete solutions depend continuously on the data of the considered problems. The stated results can be extended to multi-term time-fractional diffusion and diffusion-wave equations. One of the main features when using GDM is that their results hold 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. Some examples of schemes recovered by the GSs (Gradient Schemes) presented in this work are sketched. Among these examples, we quote the one presented in Jin et al. (2015) to approximate time fractional diffusion equations. This work extends and improves some results presented in brief or stated without proof in the notes Bradji (2018). We present some numerical tests using SUSHI introduced in Eymard et al. (2010) to support the theoretical results.