The Immersed Boundary Method (IBM) has been introduced by Peskin in the 70's in order to model and approximate fluid-structure interaction problems related to the blood flow in the heart. The original scheme makes use of finite differences for the discretization of the Navier-Stokes equations. Recently, a finite element formulation has been introduced which has the advantage of handling the presence of the solid (modeled via a Dirac delta function) in a more natural way. In this paper we review...

In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the ${L}^{2}$ norm for the sequence of discrete operators....

In this paper we consider the Maxwell resolvent operator and its finite element
approximation. In this framework it is natural the use of the edge element
spaces and to impose the divergence constraint in a weak
sense with the introduction of a Lagrange multiplier, following
an idea by Kikuchi [14].
We shall review some of the known properties for edge element
approximations and prove some new result. In particular we shall prove a
uniform convergence in the
norm for the sequence...

In this paper we discuss lowest order stabilizations of Stokes finite elements. We study the behavior of the constants in front of the error estimates in terms of the stabilization parameters and confirm with numerical tests that the bounds are sharp. Moreover, we investigate the local mass conservation properties of the considered schemes and analyze new schemes with enhanced pressure approximation, which guarantee a better local discretization of the divergence free constraint.

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