Mathematical foundations of the immersed finite element method

In this paper, we propose an immersed solid system (ISS) method to efficiently treat the fluid-structure interaction (FSI) problems. Augmenting a fluid in the moving solid domain, we introduce a volumetric force to obtain the correct dynamics for both the fluid and the structure. We further define an Euler-Lagrange mapping to describe the motion of the immersed solid. A weak formulation (WF) is then constructed and shown to be equivalent to both the FSI and the ISS under certain regularity assumptions. The weak formulation (WF) may be computed numerically by an implicit algorithm with the finite element method, and the streamline upwind/Petrov-Galerkin method. Compared with the successful immersed boundary method (IBM) by Peskin and co-workers (J Comput Phys 160:705-719, 2000; Acta Numerica 11:479-517, 2002; SIAM J Sci Stat Comput 13(6):1361-1376, 1992) the ISS method applies to more general geometries with non-uniform grids and avoids the inaccuracy in approximating the Dirac delta function.