A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Abstract

Abstract. We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamondcells, we prove a first-order convergence both in the H 1 0 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on nonconforming locally refined grids is given.Mathematics Subject Classification. 35J05, 35J25, 65N12, 65N15, 65N30.

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

“…Following Hermeline [57], Domelevo, Omnès [41] and Andreianov, Boyer, Hubert [11], we consider a DDFV mesh which is a triple T = M, M * , S described below.…”

“…We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]).…”

“…We adapt the approximations used by Eymard, Gallouët, Herbin [48] (see also [31,79,49,67]) for the nonlinear convection term, and those used by Hermeline [57,58], Domelevo, Omnès [41] and Andreianov, Boyer, Hubert [11] for the doubly nonlinear diffusion term. In 3D, we propose new DDFV schemes that possess convenient discrete duality properties.…”

“…Appendix B (see also [8,7]) is devoted to an elementary reconstruction lemma which underlies our DDFV schemes in 3D. In contrast to [41,11], we are led to penalize our DDFV schemes to ensure that the two approximations of A(u) actually converge to the same limit (see Section 3.4). The DDFV schemes constructed in Section 3 possess several convenient discrete calculus formulas that we collect in Section 4.…”

“…We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes.…”