  The immersed boundary-finite element method (IBFE) is an approach to
describing the dynamics of an elastic structure immersed in an incompressible
viscous fluid. In this formulation, there are discontinuities in the pressure
and viscous stress at fluid-structure interfaces. The standard immersed
boundary approach, which connects the Lagrangian and Eulerian variables via
integral transforms with regularized Dirac delta function kernels, smooths out
these discontinuities, which generally leads to low order accuracy. This paper
describes an approach to accurately resolve pressure discontinuities for these
types of formulations, in which the solid may undergo large deformations. Our
strategy is to decompose the physical pressure field into a sum of two
pressure-like fields, one defined on the entire computational domain, which
includes both the fluid and solid subregions, and one defined only on the solid
subregion. Each of these fields is continuous on its domain of definition,
which enables high accuracy via standard discretization methods without
sacrificing sharp resolution of the pressure discontinuity. Numerical tests
demonstrate that this method improves rates of convergence for displacements,
velocities, stresses, and pressures, as compared to the conventional IBFE
method. Further, it produces much smaller errors at reasonable numbers of
degrees of freedom. The performance of this method is tested on several cases
with analytic solutions, a nontrivial benchmark problem of incompressible solid
mechanics, and an example involving a thick, actively contracting torus.
