Title: Inf-sup stable finite-element methods for the Landau--Lifshitz--Gilbert and harmonic map heat flow equation,
Abstract: 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. \emph{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
\emph{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.