  This paper presents a numerical method for the simulation of elastic solid
materials coupled to fluid inclusions. The application is motivated by the
modeling of vascularized tissues and by problems in medical imaging which
target the estimation of effective (i.e., macroscale) material properties,
taking into account the influence of microscale dynamics, such as fluid flow in
the microvasculature. The method is based on the recently proposed Reduced
Lagrange Multipliers framework. In particular, the interface between solid and
fluid domains is not resolved within the computational mesh for the elastic
material but discretized independently, imposing the coupling condition via
non-matching Lagrange multipliers. Exploiting the multiscale properties of the
problem, the resulting Lagrange multipliers space is reduced to a
lower-dimensional characteristic set. We present the details of the stability
analysis of the resulting method considering a non-standard boundary condition
that enforces a local deformation on the solid-fluid boundary. The method is
validated with several numerical examples.
