Geodesic
Error estimate for a finite volume scheme in a geometrical multi-scale domain
Viallon, Marie-Claude1
1 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
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 529-550

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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 H 1 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 L 2 error estimate between the solution of the equation and its numerical finite volume approximation.

Reçu le :
DOI : 10.1051/m2an/2014042
Classification : 35J25, 74S10, 65N12, 65N15, 65N08
Keywords: Finite volume scheme, elliptic problem, discrete Poincaré inequality, error estimate, multi-scale domain

Viallon, Marie-Claude 1

1 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 
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Viallon, Marie-Claude. Error estimate for a finite volume scheme in a geometrical multi-scale domain. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 529-550. doi: 10.1051/m2an/2014042

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