Discretization of the coupled heat and electrical diffusion problems by the finite element and the finite volume methods

International audienceThe modelling of the heat diffusion coupled with electrical diffusion yields a nonlinear system of elliptic equations. The ohmic losses which appear as a source term in the heat diffusion equation is a nonlinear term which lies in $L^1$. A finite element scheme and a finite volume scheme are considered for the discretization of the system; in both cases, we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system

"A note on the Entropy Solutions of the Hydrodynamic Model of Traffic Flow" revisited

International audienceA central question in [Velan and Florian(2002)] is the influence of a non differentiable fundamental diagram on the solutions of the LWR model. This question is crucial because experimental observations put credit on piecewise linear fundamental diagrams (PLFD) [Leclercq(2005), Chiabaut et al.(2009)] and especially on triangular ones. In [Velan and Florian(2002)] it is claimed that, with the latter diagrams, the solution of the LWR model is unique but non-entropic. This note aims to invalidate this result. Considering a triangular fundamental diagram, we will demonstrate (i) that the weak solutions of the LWR are not unique and (ii) that the solution which is usually retained (and claimed to be unique) is in fact the unique weak entropy solution in the sense of Kruskov [Kruˇzkov(1970)]. This means that PLFD cannot be disproved on an alleged mathematical basis, since contrary to what is claimed in [Velan and Florian(2002)], the entropy criterion is indeed respected by its solutions

Convergence of linear finite elements for diffusion equations with measure data

We show here the convergence of the linear finite element approximate solutions of a diffusion equation to a weak solution, with weak regularity assumptions on the data

Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [27]. This is an extended version of the paper submitted to IMAJNA

Entropy estimates for a class of schemes for the euler equations

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero

Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study

A finite volume scheme for anisotropic diffusion problems

A new finite volume for the discretization of anisotropic diffusion problems on general unstructured meshes in any space dimension is presented. The convergence of the approximate solution and its discrete gradient is proven. The efficiency of the scheme is shown through numerical examples

Development of numerical methods for the reactive transport of chemical species in a porous media : a nonlinear conjugate gradient method

In the framework of the evaluation of nuclear waste disposal safety, the French Atomic Energy Commission (CEA) is interested in modelling the reactive transport in porous media. At a given time step, the equation system of reactive-transport can be written as a system of nonlinear coupled equations F(x) = 0. In the computational code which is presently used, this system is solved using classical sequential iterative algorithms (SIA). We are currently investigating nonlinear conjugate gradient methods to improve the resolution of the system. Indeed, the handling of the coupling is improved by numerical derivation along the descent direction. The original feature of this method is the use of an explicit formula for the descent parameter. We choose an approach involving two distinct codes, that is one code for the chemistry and one code for the transport equations

Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces

International audienceA 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

A nine point finite volume scheme for the simulation of diffusion in heterogeneous media

International audienceWe propose a cell-centred symmetric scheme which combines the advantages of MPFA (multi point ux approximation) schemes such as the L or the O scheme and of hybrid schemes: it may be used on general non conforming meshes, it yields a 9-point stencil on two-dimensional quadrangular meshes, it takes into account the heterogeneous diusion matrix, and it is coercive and convergent. The scheme relies on the use of special points, called harmonic averaging points, located at the interfaces of heterogeneity. 1