Volume 49, Number 2, March-April 2015
|Page(s)||529 - 550|
|Published online||17 March 2015|
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,
Received: 28 January 2014
Revised: 20 July 2014
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.
Mathematics Subject Classification: 35J25 / 74S10 / 65N12 / 65N15 / 65N08
Key words: Finite volume scheme / elliptic problem / discrete Poincaré inequality / error estimate / multi-scale domain
© EDP Sciences, SMAI, 2015
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