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The dynamics of the magnetic distribution in a ferromagnetic material isgoverned by the Landau-Lifshitz equation, which is a nonlinear geometricdispersive equation with a nonconvex constraint that requires the magnetizationto remain of unit length throughout the domain. In this article, we present amass-lumped finite element method for the Landau-Lifshitz equation. This methodpreserves the nonconvex constraint at each node of the finite element mesh, andis energy nonincreasing. We show that the numerical solution of our method forthe Landau-Lifshitz equation converges to a weak solution of theLandau-Lifshitz-Gilbert equation using a simple proof technique that cancelsout the product of weakly convergent sequences. Numerical tests for bothexplicit and implicit versions of the method on a unit square with periodicboundary conditions are provided for structured and unstructured meshes.

Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation