M. Ruggeri:

"Mathematical challenges in computational micromagnetics";

Talk: Seminar of applied mathematics, Pavia (invited); 2017-05-23.

The understanding of the magnetization dynamics plays an essential role in the design of many technological applications, e.g., magnetic sensors, actuators, and storage devices. The availability of reliable numerical tools to perform large-scale micromagnetic simulations of magnetic systems is therefore of fundamental importance. Time-dependent micromagnetic phenomena are usually described by the Landau-Lifshitz-Gilbert (LLG) equation. The numerical integration of the LLG equation poses several challenges: strong nonlinearities, a nonconvex pointwise constraint, an intrinsic energy law, which combines conservative and dissipative effects, as well as the presence of nonlocal field contributions, which prescribe the coupling with other partial differential equations. In this talk, we consider the numerical analysis of a class of tangent plane integrators for the LLG equation. The methods are based on equivalent reformulations of the equation in the tangent space, which are discretized by first-order finite elements and only require the solution of one linear system per time-step. The pointwise constraint is enforced at the discrete level by applying the nodal projection mapping to the computed solution at each time-step. Under appropriate assumptions, the convergence towards a weak solution of the problem is unconditional, i.e., the numerical analysis does not require to impose any CFL-type condition on the time-step size and the spatial mesh size. Numerical experiments support our theoretical findings and demonstrate the applicability of the method for the simulation of practically relevant problem sizes. This is joint work with Dirk Praetorius, Bernhard Stiftner (TU Wien), Claas Abert, and Dieter Suess (University of Vienna).

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