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Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

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Numerical Heat Transfer And Fluid Flow

We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that th...

Basic principles of Virtual Element Methods

A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.

/pdf/discretization-of-heterogeneous-and-anisotropic-diffusion-2h03y9yty3.pdf

Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the methods (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

https://hal.archives-ouvertes.fr/hal-00346077/document

A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation 
$u_t+{\rm div}({\mathbf q} f(u))-\Delta \phi(u)=0$
 by a piecewise constant function 
$u_{{\mathcal D}}$
 using a discretization 
${\mathcal D}$
 in space and time and a finite volume scheme. The convergence of 
$u_{{\mathcal D}}$
 to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on 
$u_{{\mathcal D}}$
 are used to prove the convergence, up to a subsequence, of 
$u_{{\mathcal D}}$
 to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of 
$u_{{\mathcal D}}$
 to{\it u}. Some on a model equation are shown.

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

We 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. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.

https://hal.archives-ouvertes.fr/hal-00005565/document

Robert Eymard

Papers

Finite Volume Methods

Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

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