Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation
This work provides a convergent numerical method for simulating magnetization dynamics in ferromagnetic materials, which is important for computational micromagnetics.
The authors propose a mass-lumped finite element method for the Landau-Lifshitz equation that preserves the unit-length constraint at nodes and is energy nonincreasing, and prove convergence to a weak solution of the Landau-Lifshitz-Gilbert equation.
The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.