Activités de recherche antérieures
Fictitious domain method
The fictitious domain method has been investigated for evolution problems, especially for the time dependent electromagnetic scattering problems. The main idea consists in extending the solution inside the obstacle and in working with two independent meshes, one regular and uniform for the computational domain and the other for the boundary, the boundary condition being taken in charge through the the introduction of an auxiliary function, defined only on the boundary alone. The method can be shown to converge if a particular uniform condition holds. We have undertaken to establish such a condition for Maxwell's equations. The first numerical tests in the two dimensional case have been investigated and prove the superiority of this method in terms of accuracy over the finite domain time method. The fictitious domain method is being extended in the three dimensional case.
Domain decomposition method
The principle of this method consists, as usual, in dividing the entire computational domain into smaller subdomains, in each of which the equations are supposed to be easily solved. All the difficulty lies in the way one joins the subdomains, which is achieved iteratively. The aim is to introduce a new nonlocal transmission conditions at the interfaces between the subdomains in order to improve the convergence rate of the iterative algorithm. The numerical tests using several types of nonlocal transmission conditions are done for the Helmholtz's equation in the two dimensional case . They are very promising. A parallel implentation has been done. This method will be extended in the three dimensional case.
Absorbing boundary conditions
The aim is to design a new family of higher order absorbing boundary conditionas for Maxwell's equations adapted to the use of finite elements. A new version of the transparent conditions which involves the tangential components of the electromagnetic field, has been written. The approximations of these conditions have been constructed and their stability is verified. Finally a variational formulation has been obtained, allowing us the use of finite elements for the numerical approximation. The numerical implementation of this method is under way.