Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics

Abstract

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau–Lifshitz–Gilbert equation, which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of the equation, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove linear convergence of the preconditioned GMRES algorithm. Numerical experiments underpin the theoretical findings.