Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes

摘要

summary:A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. We derive error estimates in discrete norms $\Bbb L^{\infty }(0,T;H^1_0(\Omega ))$ and ${\Cal W}^{1,\infty }(0,T;L^2(\Omega ))$, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form involved in the finite volume scheme satisfies some ellipticity condition.