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Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Noel J. Walkington — 2011

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

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Noel J. Walkington — 2011

ESAIM: Mathematical Modelling and Numerical Analysis

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Dual-mixed finite element methods for the Navier-Stokes equations

Jason S. HowellNoel J. Walkington — 2013

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

A mixed finite element method for the Navier–Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier–Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf–sup conditions are developed.

Mixed methods for the approximation of liquid crystal flows

Chun LiuNoel J. Walkington — 2002

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

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H 2 ( Ω ) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.

Lagrangian and moving mesh methods for the convection diffusion equation

Konstantinos ChrysafinosNoel J. Walkington — 2008

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington,   (2006) 2478–2499; Chrysafinos and Walkington,   (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent of...

Mixed Methods for the Approximation of Liquid Crystal Flows

Chun LiuNoel J. Walkington — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve (Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.

Approximation of Parabolic Equations Using the Wasserstein Metric

David KinderlehrerNoel J. Walkington — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature of the algorithms presented here is that derivatives do not enter into the variational principles, so, for example, discontinuous approximations may be used for approximating the heat equation. We present formulae for computing a Wasserstein metric which enters into the variational...

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