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A Fortin operator for two-dimensional Taylor-Hood elements

Richard S. Falk — 2008

ESAIM: Mathematical Modelling and Numerical Analysis

A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.

Hexahedral H(div) and H(curl) finite elements

Richard S. FalkPaolo GattoPeter Monk — 2011

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

We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using...

Basic principles of mixed Virtual Element Methods

F. BrezziRichard S. FalkL. Donatella Marini — 2014

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

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of (div)-conforming vector fields (or, more generally, of ( − 1) − ). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making...

Hexahedral (div) and (curl) finite elements

Richard S. FalkPaolo GattoPeter Monk — 2011

ESAIM: Mathematical Modelling and Numerical Analysis

We study the approximation properties of some finite element subspaces of (div;Ω) and (;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral (div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using the...

Analysis of a one-dimensional variational model of the equilibrium shapel of a deformable crystal

Eric BonnetierRichard S. FalkMichael A. Grinfeld — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The equilibrium configurations of a one-dimensional variational model that combines terms expressing the bulk energy of a deformable crystal and its surface energy are studied. After elimination of the displacement, the problem reduces to the minimization of a nonconvex and nonlocal functional of a single function, the thickness. Depending on a parameter which strengthens one of the terms comprising the energy at the expense of the other, it is shown that this functional may have a stable absolute...

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