Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics
For computational micromagnetics, this provides a provably convergent and efficient solver for a key numerical scheme, though it is an incremental improvement over existing methods.
The paper develops an efficient solution strategy for constrained linear systems arising from the tangent plane scheme for the Landau-Lifshitz-Gilbert equation, using Householder reflections and preconditioners that yield linear convergence in GMRES, with time-step-independent preconditioners.
The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation (LLG), which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of LLG, 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 that the preconditioned GMRES algorithm leads to linear convergence. Numerical experiments underpin the theoretical findings.