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C.-M. Pfeiler, D. Praetorius, M. Ruggeri, B. Stiftner:
"A convergent second-order implicit-explicit tangent plane scheme for the Landau-Lifshitz-Gilbert equation";
Talk: RMMM 8 - Reliable Methods of Mathematical Modeling, Berlin; 2017-07-31 - 2017-08-03.

We consider the numerical approximation of the Landau-Lifshitz-Gilbert (LLG) equation, which describes the dynamics of the magnetization in ferromagnetic materials. The numerical integration of the LLG equation poses several challenges: strong nonlinearities, a nonconvex pointwise constraint, an intrinsic energy law, and the presence of nonlocal field contributions, which prescribe the coupling with other partial differential equations (PDEs). We propose a tangent plane scheme in which all the lower-order contributions, e.g., the expensive-to-compute stray field, are treated explicitly in time by using an Adams-Bashforth approach. The numerical integrator is computationally attractive in the sense that, after the first time-step, only one linear system has to be solved per time-step. Despite this modification, the resulting scheme still fulfills a formal second-order convergence in time. Under appropriate assumptions, the convergence towards a weak solution of the problem is unconditional, i.e., no CFL-type condition on the time-step size and the spatial mesh size is needed for the stability of the scheme. One particular focus is on the efficient treatment of coupled systems, e.g., the coupling of the LLG equation with the eddy current equation, for which we show that decoupling the time integration of the two PDEs is very attractive in terms of computational cost and still leads to time-marching algorithms that are (unconditionally) convergent and of second-order in time. Numerical experiments underpin our theoretical findings and demonstrate the applicability of the method for the simulation of practically relevant problem sizes.