Stability results for the finite element approach to the immersed boundary method

The immersed boundary method is both a mathematical formulation and a numerical method for the study of fluid structure interactions. Many numerical schemes have been introduced to reduce the difficulties related to the non-linear coupling between the structure and the fluid evolution; however numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates for the variational formulation of the immersed boundary method.A two dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier-Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution (BE/BE), and we present a computable CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly (FE/BE)