A mixed finite volume scheme for anisotropic diffusion problems on any grid

Abstract

International audienceWe present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. In the case of simplicial meshes, the approximate solution is shown to converge to the continuous ones as the size of the mesh tends to 0, and an error estimate is given. In the general case, we propose a slightly modified scheme for which we again prove the convergence, and give an error estimate. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

“…Eymard et al [36], Aavatsmark et al [1], or Droniou and Eymard [32], mimetic finite difference; cf. Brezzi et al [21], covolume; cf.…”

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

“…A major drawback is their lack of stability in some configurations. Two ways of overcoming this difficulty by designing discretizations based on the variational formulation of the problem and featuring cell-and face-unknowns have been proposed by Brezzi, Lipnikov and coworkers [8,9] (Mimetic Finite Difference methods) and by Droniou and Eymard [20] (Mixed/Hybrid Finite Volume methods). In this context, Eymard, Gallouët and Herbin [24] have shown that face unknowns can be selectively used as Lagrange multipliers to enforce flux continuity, or eliminated using a consistent interpolator (SUSHI scheme).…”

“…This topic is not addressed in detail herein since our focus is mainly on implementation. For a comprehensive discussion we refer to Brezzi, Lipnikov and coworkers [8,9], Droniou and Eymard [20], Eymard, Gallouët, and Herbin [24], Di Pietro and Ern [17, Chap. 1] and Di Pietro [15].…”

“…The SUSHI method with hybrid unknowns reads 19) and ∇ h broken gradient on P h . Alternatively, one can obtain a cell centered version by setting 20) and T g h defined by (2.11). This variant coincides with the version proposed in [24] for homogeneous κ, but it has the advantage to reproduce piecewise affine solutions to (2.3) on T h when κ is heterogeneous.…”

The G method for heterogeneous anisotropic diffusion on general meshes

“…In [25], the equivalence of multipoints methods with the lowest-order Mixed Finite Element method on matching triangular grids has been pointed out, and local coercivity conditions have been proposed. Other relatively inexpensive methods that deserve being mentioned are those developed in the Mimetic Finite Difference framework of [14][15][16], as well as the Hybrid Finite Volume scheme of [24] or the Mixed Finite Volume scheme of [19]. The analogies among the three classes of methods have been recently pointed out in [20].…”