Résumé : |
In the context of electromagnetic wave propagation, we wish to adress
the scattering problem from perfectly conducting thin wires. For
numerical simulations, assuming
their thickness to be much smaller than the wavelength of the
incident field, it is not possible to take these obstacles into
account without encountering problems of numerical locking.
The Holland model (cf. \cite{Hol}), widely used in finite difference schemes, provides
a pragmatic solution to this problem, by modifying the numerical
scheme on vertices located in the neighbourhood of the wires. So far
this model has not received any real theoretical justification, and
involves a parameter, named lineic inductance,
which is to be chosen on the basis of heuristic considerations.
We are interested in the simplified problem of a bidimensional acoustic wave
propagation in a medium including a small obstacle with homogeneous
Dirichlet boundary condition. We present a numerical scheme suitable
for finite elements that does not suffer from numerical locking, and
takes the presence of the small obstacle into account. It is based on
a combination between the fictitious domain method and matched
asymptotic expansions. This results into a systematic generalization
to the Holland model including an automatic computation of the
lineic inductance. Our analysis leads to the first (to our knowledge)
justification of this type of model. Even if we shall reduce our
presentation to the 2D time harmonic case, it appears
that this approach can be generalized to the 3D time dependent case
(cf. \cite{Rogier},\cite{Fedoryuk},\cite{Art}). |