Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P1 function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L2 norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to H1(Ω).
@article{M2AN_2011__45_4_627_0,
author = {Omnes, Pascal},
title = {On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {45},
year = {2011},
pages = {627-650},
doi = {10.1051/m2an/2010068},
mrnumber = {2804653},
zbl = {1269.65109},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2011__45_4_627_0}
}
Omnes, Pascal. On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 627-650. doi : 10.1051/m2an/2010068. http://gdmltest.u-ga.fr/item/M2AN_2011__45_4_627_0/
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