Error estimate for a finite volume scheme in a geometrical multi-scale domain

Universitéde Lyon, UMR CNRS 5208, Université Jean Monnet, Institut Camille Jordan, Faculté des Sciences et Techniques, 23 rue Docteur Paul Michelon, 42023 Saint-Etienne cedex 2, France
jf.us.gerbeau@gmail.com

Received: 28 January 2014
Revised: 20 July 2014

Abstract

We study a finite volume scheme, introduced in a previous paper [G.P. Panasenko and M.-C. Viallon, Math. Meth. Appl. Sci. 36 (2013) 1892–1917], to solve an elliptic linear partial differential equation in a rod structure. The rod-structure is two-dimensional (2D) and consists of a central node and several outgoing branches. The branches are assumed to be one-dimensional (1D). So the domain is partially 1D, and partially 2D. We call such a structure a geometrical multi-scale domain. We establish a discrete Poincaré inequality in terms of a specific H¹ norm defined on this geometrical multi-scale 1D-2D domain, that is valid for functions that satisfy a Dirichlet condition on the boundary of the 1D part of the domain and a Neumann condition on the boundary of the 2D part of the domain. We derive an L² error estimate between the solution of the equation and its numerical finite volume approximation.