Applied Math Seminar
Winter Quarter 2009
12:00 p.m.
Sloan Mathematics Corner
Building 380, Room 380-W

Friday, February 27, 2009
SPECIAL: 12:00p in room 380-W

Sonia Fliss
INRIA
Rocquencourt, France

In this talk, I will present a strategy developed jointly with Patrick Joly for handling 2D time-harmonic wave propagation in locally perturbed infinite periodic media. The main difficuty lies in the reduction of the effective numerical computations to a bounded region enclosing the perturbation. Our objective is to extend the approach by Dirichlet-to-Neumann operators, well known in the case of homogeneous media ( as non local transparent boundary conditions). The new difficulty is that this DtN operator can no longer be determined explicitly and has to be computed numerically. The general idea is to take advantage of the periodic structure of the problem to design a method for the construction of exact DtN conditions by solving only problems posed on one periodicity cell.

For simplicity, we shall restrict ourselves to the case where the periodicity cell is a square and presents at least two symmetries, which is often the case in the applications. We want to restrict the computations to a square whose side is a multiple of the size of the periodicity cell. We show that the DtN operator can be characterized through the solution of local PDE cell problems, the use of the Floquet-Bloch transform and the solution of operator-valued quadratic or linear equations.

The theoretical aspects of the problem and corresponding numerical issues will be addressed in the talk. The non standard aspects of this procedure will be emphasized and numerical results demonstrating the efficiency of the method will be presented.