In this paper, we are interested in the modelling and the finite element
approximation of a petroleum reservoir, in axisymmetric form. The flow in the
porous medium is governed by the Darcy-Forchheimer equation coupled with a
rather exhaustive energy equation. The semi-discretized problem is put under a
mixed variational formulation, whose approximation is achieved by means of
conservative Raviart-Thomas elements for the fluxes and of piecewise constant
elements for the pressure and the temperature....
A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element...
A general setting is proposed for the mixed finite element approximations of
elliptic differential problems involving a unilateral boundary condition. The
treatment covers the Signorini problem as well as the unilateral contact
problem with or without friction. Existence, uniqueness for both the
continuous and the discrete problem as well as error estimates are established
in a general framework. As an application, the approximation of the Signorini
problem by the lowest order mixed finite element...
We present here a series of works which aims at describing geophysical flows in the equatorial zone, taking into account the dominating influence of the earth rotation. We actually proceed by successive approximations computing for each model the response of the fluid to the strong Coriolis penalisation. The main difficulty is due to the spatial variations of the Coriolis acceleration : in particular, as it vanishes at the equator, fast oscillations are trapped in a thin strip of latitudes.
There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of...
In the second part of the paper, we compare the solutions produced in the framework of the conference “Mathematical and numerical aspects of low Mach number flows” organized by INRIA and MAB in Porquerolles, June 2004, to the reference solutions described in Part 1. We make some recommendations on how to produce good quality solutions, and list a number of pitfalls to be avoided.
In the second part of the paper, we compare the solutions produced
in the framework of the conference “Mathematical and numerical
aspects of low Mach number flows” organized by INRIA and MAB in
Porquerolles, June 2004, to the reference solutions described in
Part 1. We make some recommendations on how to produce good
quality solutions, and list a number of pitfalls to be avoided.
There are very few reference solutions in the literature on
non-Boussinesq natural convection flows. We propose here a test
case problem which extends the well-known De Vahl Davis
differentially heated square cavity problem to the case of large
temperature differences for which the Boussinesq approximation is
no longer valid. The paper is split in two parts: in this first
part, we propose as yet unpublished reference solutions for cases
characterized by a non-dimensional temperature difference...
For the Stokes problem in a two- or three-dimensional
bounded domain, we propose a new mixed finite element discretization which relies on
a nonconforming approximation of the velocity and a more accurate approximation of the
pressure. We prove that the velocity and pressure discrete spaces are compatible, in the
sense that they satisfy an inf-sup condition of Babuška and Brezzi type, and we
derive some error estimates.