Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The numerical resolution of partial differential equations in
domains with an intricate, a priori unknown or deforming over time
geometry, generally induce difficulties related to mesh generation.
Indeed, solving such problems requires, classically, the generation of
conforming meshes in order to impose boundary or interface conditions in
a strong way. The immersed boundary methods were introduced in order to
overcome this limitation and to be able to consider non-conforming, even
structured or cartesian meshes. In return, special treatment is required
at the domain boundary or interface.

We are interested in an optimal control non-conforming mesh method,
initially proposed for a Poisson-Dirichlet problem with constant
coefficient in intricate geometries. In this context, the computation
domain is extended to a larger, and easier to mesh, domain as is often
the case for fictitious domain methods. The main difficulty is then to
take into account the boundary conditions since the mesh does not fit
the boundary. The particularity of the present method is to introduce a
source term in the fictitious region of the domain in order to impose
the boundary conditions. This source term, called “control” in the
sequel, is chosen to solve a least squares problem involving the
boundary conditions.

Throughout the presentation, we will prove the validity of the method
applied to a Poisson-Dirichlet problem and present a recent convergence
result for the solution of the associated finite element problem. We
will also show that this method can be extended to systems of coupled
elliptic partial differential equations, such as a fluid-structure
interaction problem with rigid bodies immersed in a Stokes flow or more
general elliptic transmission problems (Poisson, Stokes, linearized
elasticity coupled with Stokes).