An analysis of a second-order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes

摘要

The present work is an extension of our previous work (Bradji and Fuhrmann (2011) [4]) which dealt with error analysis of a finite volume scheme of first order (both in time and space) for parabolic equations on general nonconforming multidimensional spatial meshes introduced recently in Eymard et al. (2010) [12]. We aim in this paper to get a higher-order time accurate scheme for a finite volume method for parabolic equations using the same class of spatial generic meshes stated above.We derive a finite volume scheme approximating the heat equation, as a model for parabolic equations, in which the discretization in time is performed using the Crank–Nicolson method.We derive an a priori estimate for the discrete problem and we prove that the error estimate of the finite volume scheme is of order two in time and it is of optimal order in space.The error estimate is analysed in several norms which allow us to derive approximations for the exact solution and its first derivatives (both spatial and temporal) whose the convergence order is two in time and it is optimal in space.We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is k2+hD, where hD (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption u∈C3([0,T];C2(Ω¯)) for the exact solution u.The proof of these error estimates is based essentially on a new a priori estimate for the discrete problem and a comparison between the finite volume approximate solution and an auxiliary finite volume approximation.