Numerical methods for the Landau-Lifshitz equation in micromagnetics : the mimetic finite difference method and the mass-lumped finite element method
Micromagnetics is a continuum theory describing magnetization patterns inside ferromagnetic media. The dynamics of a ferromagnetic material are governed by the Landau-Lifshitz equation. This equation is highly nonlinear, has a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain, which pose interesting challenges in developing numerical methods. In this thesis, we first present a low order mimetic finite difference method for the Landau-Lifshitz equation, that works on general polytopal meshes on general geometries, preserves non-convex constraint, is energy (exchange) decreasing, requires only a linear solver at each time step and is easily applicable to the limiting cases. Secondly, we present a high order mimetic finite difference method for the Landau-Lifshitz equation which is third order in space and second order in time. In fact, it can be arbitrarily high order in space. This method works on general polytopal meshes, and preserves the non-convex constraint in a certain sense. Lastly, we present a new class of convergent mass-lumped finite element methods to solve a weak formulation of the Landau-Lifshitz equation. The scheme preserves a non-convex constraint, requires only a linear solver at each time step and is easily applicable to the limiting cases. We provide a rigorous convergence proof that the numerical solution of our finite element method for the Landau-Lifshitz equation converges weakly to a weak solution of the Landau-Lifshitz-Gilbert equation.