A finite element approach to the immersed boundary method

The immersed boundary method was introduced by Peskin in [31] to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. The basic idea is to consider the structure as a part of the fluid where additional forces are applied and additional mass is localized. The forces exerted by the structure on the fluid are taken into account as a source term in the Navier-Stokes equations and are mathematically described as a Dirac delta function lying along the immersed structure. In this paper we first review on various ways of modeling the elastic forces in different physical situations. Then we focus on the discretization of the immersed boundary method by means of finite elements which can handle the Dirac delta function variationally avoiding the introduction of its regularization. Practical computational aspects are described and some preliminary numerical experiment in two dimensions are reported.