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

Pascal Omnes (1, 2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 SFME - Service Fluide numériques, Modélisation et Etudes

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~$H^1(\Omega)$.

Keywords: convergence; Voronoi meshes; Laplace equation; Delaunay meshes; finite volume method; error estimates

Domains: Numerical Analysis [math.NA]

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, 2011, 45 (4), pp.627--650. ⟨10.1051/m2an/2010068⟩. ⟨cea-00430941v2⟩