Title: An immersed boundary method based on the resolution of an optimal control problem
Abstract
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 one easier to mesh, 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-interaction problem with rigid bodies immersed in a Stokes flow or more general elliptic transmission problems (Poisson, Stokes, linearized elasticity coupled with Stokes).
