Title
Inf-Sup Stable Finite Element Methods for the Landau-Lifshitz-Gilbert and Harmonic Map Heat Flow Equations.
Abstract
In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau-Lifshitz-Gilbert equations, 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. 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. These extra elaborations are needed due to the fact that any crude time integration either does not keep the unit length at the nodes or does not satisfy an energy law. We will carry out a linear Euler-like time-stepping method and a nonlinear Crank-Nicolson-like time-stepping method, which satisfy the desired properties. The Crank-Nicolson method is solved by using the idea of linearization for the Euler method as iterations.
Year
DOI
Venue
2017
10.1137/17M1116799
SIAM JOURNAL ON NUMERICAL ANALYSIS
Keywords
Field
DocType
finite element approximation,inf-sup conditions,Landau-Lifshitz-Gilbert equation,harmonic map heat flow equation
Mathematical optimization,Harmonic map,Saddle point,Mathematical analysis,Lagrange multiplier,Landau–Lifshitz–Gilbert equation,A priori and a posteriori,Finite element method,Mathematics,Unit sphere,Mixed finite element method
Journal
Volume
Issue
ISSN
55
6
0036-1429
Citations 
PageRank 
References 
0
0.34
0
Authors
2
Name
Order
Citations
PageRank
Juan Vicente Gutiérrez-Santacreu1174.58
Marco Restelli200.34