A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

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 method (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, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. 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

Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the Brezzi-Pitk\"aranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the $L^2$-convergence of the components of the velocity, and, in the steady case, the weak $L^2$-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov's theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.Comment: submitted September 0

A collocated finite volume scheme to solve free convection for general non-conforming grids

We present a new collocated numerical scheme for the approximation of the Navier-Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity-pressure unknowns. This scheme is based on a recent scheme for the diffusion terms. Stability properties are drawn from particular choices for the pressure gradient and the non-linear terms. Numerical results show the accuracy of the scheme on irregular grids

Convergence in C([0,T];L2(Ω))C(\lbrack0,T\rbrack;L^2(\Omega)) of weak solutions to perturbed doubly degenerate parabolic equations

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to perturbations of the data. Beginning with a proof of temporally-uniform, spatially-weak convergence, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates on the solution. The double degeneracy --- shown to be equivalent to a maximal monotone operator framework --- is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.Comment: J. Differential Equations, 201

A unified analysis of elliptic problems with various boundary conditions and their approximation

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincar{\'e} inequalities and the surjectivity of the divergence operator in appropriate spaces

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

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. 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

L’accompagnement à l’observance thérapeutique des personnes toxicomanes sous traitement de substitution en situation de précarité

Contexte : Évaluation d’un projet expérimental sur l’accompagnement à l’observance du traitement de substitution des personnes toxicomanes en situation de précarité. Méthode : Analyse linguistique statistique et analyse compréhensive du récit des pratiques de 13 professionnels d’un centre de soins et de dix personnes sous traitement de substitution. Discussion : L’accompagnement à l’observance thérapeutique permettant d’allier qualité de vie et meilleur état de santé est devenu un objectif de santé publique. Si la compliance n’est plus considérée aujourd’hui comme un objectif de l’éducation en santé, elle n’est pas pour autant exclue des pratiques. Les pratiques d’accompagnement déclarées s’inscrivent autant dans un modèle de guidage de l’action et de l’instruction que dans un modèle privilégiant le cheminement avec le sujet dans un processus d’observance/non-observance tentant d’allier les risques en santé et la qualité de vie. Les professionnels déclarent développer des stratégies, des astuces, pour accompagner ces personnes dans un équilibre de santé et de qualité de vie. La multiplicité des obstacles à l’observance et leur imbrication multidirectionnelle invite à considérer la santé dans son approche biopsychosociale d’un sujet et d’un groupe autonome et capable de prendre des décisions de santé lui permettant d’exister dans un environnement en évolution.Context: The evaluation of an experimental counselling project carried out with socially unstable drug-addicts to monitor the compliance with their substitution treatment. Method: Statistical analysis of linguistics and comprehensive analysis of the practice of 13 professionals and the accounts of 10 subjects in a treatment centre. Discussion: It has become a goal in public health to tie together quality of life and improved health through counselling with therapeutic compliance. Compliance is no longer considered a priority in health education; however, it has not been completely discarded. Two models were described in the counselling practice: an action and learning guidance model, and a model which accompanies the subject in the compliance/non-compliance process by looking for the ties between health risks and quality of life. The professionals state that they develop strategies to support these individuals in finding a balanced health and quality of life. The multiplicity of obstacles to compliance and their multidirectional overlap suggest that health should be considered in a bio-psycho-social approach to a person and independent group, capable of making health decisions that allow them to exist in an evolving environment.Contexto: evaluación de un proyecto experimental sobre el acompañamiento de personas toxicómanas en situación precaria en la observancia del tratamiento de sustitución. Método: análisis lingüístico estadístico y análisis comprensivo del relato de las prácticas de trece profesionales de un centro de tratamiento y de diez personas en tratamiento de sustitución. Discusión: el acompañamiento en la observancia terapéutica que permite unir la calidad de vida y el mejoramiento del estado de salud se ha convertido en un objetivo de salud pública. Si el cumplimiento ya no se considera hoy un objetivo de educación sanitaria, el mismo no está sin embargo excluido de las prácticas. Las prácticas de acompañamiento declaradas se inscriben tanto en un modelo de guía de la acción y de la instrucción como en un modelo que privilegia el progreso con el sujeto en un proceso de observancia o no-observancia que trata de vincular los riesgos para la salud y la calidad de vida. Los profesionales de la salud declaran desarrollar estrategias y astucias para acompañar a estas personas en un equilibrio de salud y calidad de vida. La multiplicidad de los obstáculos que se presentan a la observancia y su imbricación multidireccional invitan a considerar, en su enfoque bio-psicosocial, la salud de un sujeto y de un grupo autónomo y capaz de tomar, en materia de salud, las decisiones que le permiten existir en un medio en evolución

Unified convergence analysis of numerical schemes for a miscible displacement problem

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $L^\infty(0,T; L^2(\Omega))$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion

A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension

Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality condition. This discrete gradient is shown to satisfy a strong convergence property on the interpolation of regular functions, and a weak one on functions bounded for a discrete $H^1$ norm. To highlight the importance of both properties, the convergence of the finite volume scheme on a homogeneous Dirichlet problem with full diffusion matrix is proven, and an error estimate is provided. Numerical tests show the actual accuracy of the method