Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem

Abstract

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate of order h, assuming only the W 2,p (for p > 2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in [12]. Some new difficulties arise here, due to the weak regularity of the solution, and the necessity to approximate the entire gradient, and not only its normal component, as in [12]. Résumé. Dans cet article, onétudie une classe de schémas volumes finis sur des maillages stucturés généraux, pour un problème linéaire de convection diffusion. La convection est approchée par un schéma décentré amont, et la diffusion par un schéma dit "des cellules diamants" [4]. On démontre une estimation d'erreur d'ordre h pour une solution continue dans W 2,p (p > 2), sur des maillages de quadrangles. La démonstration est une généralisation des idées de [12]. Les nouvelles difficultés sont la régularité plus faible de la solution exacte et la nécessité de construire une approximation du gradient et pas seulement de sa composante normale aux interfaces.

Reduced basis method for finite volume approximations of parametrized linear evolution equations

“…For the convective part usual choices are monotone fluxes such as the Lax-Friedrichs, Enquist-Osher or the Godunov flux [5,20]. For the diffusive flux simple difference approximations of the gradient can be performed [20,27], occasionally involving harmonic weighting of cell-averaged diffusivities [10,17], or more general gradient reconstruction techniques [9,12].…”

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…The key feature of the 2D schemes of [3,56,12,10,8] (see also [40]) lies in the fact that the fluxes across interfaces are reconstructed "manually". The approaches of Droniou and Eymard [47,42] and those of the HVF, SUCCES and SUSHI schemes of Eymard, Gallouët and Herbin [50,51,53,52] are different; they rely on introducing additional unknowns (either for the fluxes, or for the values on some of the edges) and on careful penalization of the finite differences.…”

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

“…Finally, DDFV schemes use a four-point gradient formula defined in [15] on the so-called "diamond cells", whose diagonals are the primal and associated dual edges. Such schemes for the Laplace equation have been shown to converge in [19] on very general meshes, with first-order accuracy in the broken energy norm, as well as in the discrete L 2 (Ω) norm, provided the solution of the Laplace equation belongs to H 2 (Ω).…”

“…First, the principle of the so-called "cell-centered" schemes is to associate discrete unknowns with the cells of the mesh and to integrate the Laplace equation on each cell. Among various approaches, (which have been developed mainly for anisotropic diffusion, but which may of course be applied to the Laplace equation), we may cite [1,2,6,9,15,25,[29][30][31]. The principle of the second family, the so-called "vertex-centered" schemes, is to associate discrete unknowns with the vertices of the primal mesh, and then integrate the Laplace equation on the cells of a dual mesh, centered on the vertices [4,5,10,11,24,35].…”