# Inf-sup stable finite-element methods for the Landau--Lifshitz--Gilbert and harmonic map heat flow equation

## TL;DR

This paper develops and analyzes an inf-sup stable finite element method for the harmonic map heat and Landau--Lifshitz--Gilbert equations, ensuring the unit sphere constraint at nodes through a saddle point approach.

## Contribution

It introduces a unified saddle point finite element formulation with proven inf-sup stability for both equations, including energy estimates and handling of time integration complexities.

## Key findings

- Proved inf-sup condition for the Lagrange multiplier.
- Established a priori energy estimates for the methods.
- Implemented Euler and Crank--Nicolson time-stepping schemes.

## 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.