Numerical Stability of the Finite Element Immersed Boundary Method

The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully non-linearly coupled formulation for the study of fluid structure interactions. As it is common in these cases, many numerical methods 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 of 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, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly. We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.