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The Immersed Boundary Method for Fluid-Structure Interactions: Mathematical Formulation and Numerical

Daniele Boffi — 2012

Bollettino dell'Unione Matematica Italiana

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...

Edge finite elements for the approximation of Maxwell resolvent operator

Daniele BoffiLucia Gastaldi — 2002

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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....

Edge finite elements for the approximation of Maxwell resolvent operator

Daniele BoffiLucia Gastaldi — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Stabilized Stokes Elements and Local Mass Conservation

Daniele BoffiNicola CavalliniFrancesca GardiniLucia Gastaldi — 2012

Bollettino dell'Unione Matematica Italiana

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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