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We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases,… (More)

- Franck Boyer, Florence Hubert
- SIAM J. Numerical Analysis
- 2008

In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the Discrete Duality Finite Volume framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent… (More)

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic… (More)

- Dominique Barbolosi, Assia Benabdallah, Florence Hubert, Federico Verga
- Mathematical biosciences
- 2009

In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies are systemic treatments which aim at the elimination of the micrometastases produced by a primitive tumour. The efficiency of chemotherapies closely depends on the protocols of administration. Mathematical modeling is an invaluable tool to help in evaluating the… (More)

This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh’s interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes… (More)

- René Cautrès, Raphaèle Herbin, Florence Hubert
- 2007

Abstract We propose a new finite volume scheme for convection diffusion equation on non matching grids. We give error estimates for H solutions of the continuous problem. We then present a finite volume version of an adaptation of the Schwarz algorithm due to P.L. Lions, and prove, for a fixed mesh, its convergence towards the finite volume scheme on the… (More)

- Yves Coudière, Florence Hubert
- SIAM J. Scientific Computing
- 2011

Discrete Duality Finite Volume (DDFV) schemes have recently been developed in 2D to approximate nonlinear diffusion problems on general meshes. In this paper, a 3D extension of these schemes is proposed. The construction of this extension is detailed and its main properties are proved: a priori bounds, well-posedness and error estimates. The practical… (More)

- Niklas Hartung, Séverine Mollard, +8 authors Florence Hubert
- Cancer research
- 2014

Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatment strategies. In this respect, early detection of occult metastasis invisible to current imaging methods would have a major impact on best care and long-term survival. Mathematical models that describe metastatic spreading might estimate the risk of metastasis… (More)

- Boris P. Andreianov, Mostafa Bendahmane, Florence Hubert
- Comput. Meth. in Appl. Math.
- 2013

We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discreteW 1,p compactness, discrete compactness in space and in time) for the so-called Discrete Duality (DDFV) Finite Volume schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in… (More)

- Franck Boyer, Florence Hubert, Jérôme Le Rousseau
- SIAM J. Control and Optimization
- 2010