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On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Abstract : 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 $P^1$ 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~$L^2$ norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to~$H^1(\Omega)$.
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Pascal Omnes. 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, EDP Sciences, 2011, 45 (4), pp.627--650. ⟨10.1051/m2an/2010068⟩. ⟨cea-00430941v2⟩

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