Inf-sup stable finite-element methods for the Landau--Lifshitz--Gilbert and harmonic map heat flow equation
This work provides a rigorous numerical analysis framework for simulating micromagnetics and harmonic maps, addressing the challenge of enforcing the unit sphere constraint in finite element discretizations.
The authors propose a unified saddle-point finite element method for the harmonic map heat flow and Landau-Lifshitz-Gilbert equations, proving inf-sup stability and a priori energy estimates. The method handles the unit sphere constraint via Lagrange multipliers, with linear Euler and nonlinear Crank-Nicolson time-stepping schemes analyzed.
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.