Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint a...

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Autores: Gutiérrez Santacreu, Juan Vicente, Restelli, Marco
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2017
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/89745
Acceso en línea:https://hdl.handle.net/11441/89745
https://doi.org/10.1137/17M1116799
Access Level:acceso abierto
Palabra clave:Finite-element approximation
Inf-sup conditions
Landau–Lifshitz–Gilbert equation
Harmonic map heat flow equation
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spelling Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations Gutiérrez Santacreu, Juan Vicente Restelli, Marco Finite-element approximation Inf-sup conditions Landau–Lifshitz–Gilbert equation Harmonic map heat flow equation In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank–Nicolson method. The latter is solved by using the former as a non-linear solver. Ministerio de Economía y Competitividad MTM2015-69875-P SIAM: Society for Industrial and Applied Mathematics https://hdl.handle.net/11441/89745 https://doi.org/10.1137/17M1116799
title Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
spellingShingle Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
Gutiérrez Santacreu, Juan Vicente
Finite-element approximation
Inf-sup conditions
Landau–Lifshitz–Gilbert equation
Harmonic map heat flow equation
title_short Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
title_full Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
title_fullStr Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
title_full_unstemmed Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
title_sort Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations
author Gutiérrez Santacreu, Juan Vicente
author_facet Gutiérrez Santacreu, Juan Vicente
Restelli, Marco
author_role author
author2 Restelli, Marco
author2_role author
topic Finite-element approximation
Inf-sup conditions
Landau–Lifshitz–Gilbert equation
Harmonic map heat flow equation
topic_facet Finite-element approximation
Inf-sup conditions
Landau–Lifshitz–Gilbert equation
Harmonic map heat flow equation
description In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank–Nicolson method. The latter is solved by using the former as a non-linear solver.
publishDate 2017
format article
status_str submittedVersion
url https://hdl.handle.net/11441/89745
https://doi.org/10.1137/17M1116799
eu_rights_str_mv openAccess
publisher SIAM: Society for Industrial and Applied Mathematics
institution Universidad de Sevilla (US)
collection idUS. Depósito de Investigación de la Universidad de Sevilla
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
instname_str Universidad de Sevilla (US)
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publishDateSort 2017
author_browse Gutiérrez Santacreu, Juan Vicente
Restelli, Marco
publisherStr SIAM: Society for Industrial and Applied Mathematics
score 6,8972664

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