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

Abstract

Abstract. 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. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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

“…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). Additional convergence results for anisotropic and/or non linear diffusion and/or discontinuous coefficients may be found in [3,8], see also [33]. For such schemes, an almost second-order accuracy result in the L 2 norm was shown (see [19], Thm.…”

“…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]. More recently, a third family of schemes has emerged, which combines the previous two approaches, since these schemes associate unknowns with both the cells and the vertices of the mesh, and integrate the Laplace equation on both the cells of the primal and dual meshes [3,13,16,18,19,26,27,33]. The originality of these schemes is that they work well on all kind of meshes, including very distorted, degenerating, or highly nonconforming meshes (see the numerical tests in [19]).…”

Analysis on general meshes of a discrete duality finite volume method for subsurface flow problems

“…In a general case where L 2 ( ) is replaced by L P ( ), with 1 < P < +∞, the proof can, for instance, be found in [3] (see also [4] Proof The eigenvalues λ of the symmetric positive definite matrix K GH satisfy to the so-called characteristic equation associated with K GH , i.e.,…”

“…Note that this second formulation has been given focused attention by some mathematicians as Andreianov, Boyer, and Hubert who have greatly contributed to its mathematical development. Indeed, key ideas involved in the pressure gradient reconstruction approach have been generalized by these authors (see [3]) to nonlinear operators of Leray-Lions type. Motivated by the possibility of increasing the order of convergence of the pressure gradient reconstruction method for nonlinear operators, Boyer and Hubert have proposed in [9] the so-called modified DDFV.…”

“…The theoretical analysis strategy of the flux-based DDFV developed in [30,32] on a rectangular mesh follows closely ideas from [14]. Very different ingredients are used in [3] and [9] for getting either the numerical scheme or the error estimates.…”