A finite element approach for the immersed boundary method

We consider a modification of the immersed boundary method which makes use of a finite element spatial discretization. We describe the method for a two-dimensional model problem and justify its variational formulation. We provide a preliminary analysis of the continuous problem in a one-dimensional setting using a fixed point theorem and a compactness argument. Finally, we report on some numerical tests which demonstrate the stability and robustness of the algorithm.