  The immersed boundary method is an approach to fluid-structure interaction
that uses a Lagrangian description of the structural deformations, stresses,
and forces along with an Eulerian description of the momentum, viscosity, and
incompressibility of the fluid-structure system. The original immersed boundary
methods described immersed elastic structures using systems of flexible fibers,
and even now, most immersed boundary methods still require Lagrangian meshes
that are finer than the Eulerian grid. This work introduces a coupling scheme
for the immersed boundary method to link the Lagrangian and Eulerian variables
that facilitates independent spatial discretizations for the structure and
background grid. This approach employs a finite element discretization of the
structure while retaining a finite difference scheme for the Eulerian
variables. We apply this method to benchmark problems involving elastic, rigid,
and actively contracting structures, including an idealized model of the left
ventricle of the heart. Our tests include cases in which, for a fixed Eulerian
grid spacing, coarser Lagrangian structural meshes yield discretization errors
that are as much as several orders of magnitude smaller than errors obtained
using finer structural meshes. The Lagrangian-Eulerian coupling approach
developed in this work enables the effective use of these coarse structural
meshes with the immersed boundary method. This work also contrasts two
different weak forms of the equations, one of which is demonstrated to be more
effective for the coarse structural discretizations facilitated by our coupling
approach.
