3 research outputs found
Geometric control condition for the wave equation with a time-dependent observation domain
We characterize the observability property (and, by duality, the
controllability and the stabilization) of the wave equation on a Riemannian
manifold with or without boundary, where the observation (or control)
domain is time-varying. We provide a condition ensuring observability, in terms
of propagating bicharacteristics. This condition extends the well-known
geometric control condition established for fixed observation domains. As one
of the consequences, we prove that it is always possible to find a
time-dependent observation domain of arbitrarily small measure for which the
observability property holds. From a practical point of view, this means that
it is possible to reconstruct the solutions of the wave equation with only few
sensors (in the Lebesgue measure sense), at the price of moving the sensors in
the domain in an adequate way.We provide several illustrating examples, in
which the observationdomain is the rigid displacement in of a fixed
domain, withspeed showing that the observability property depends both on
and on the wave speed. Despite the apparent simplicity of some of
ourexamples, the observability property can depend on nontrivial
arithmeticconsiderations
Mixed finite element approximation for a coupled petroleum reservoir model
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. The discrete problem thus
obtained is well-posed and a posteriori error estimates are also established.
Numerical tests are presented validating the developed code