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C.-M. Pfeiler:
"Numerical analysis and efficient simulation of micromagnetic phenomena";
Supervisor, Reviewer: D. Praetorius, S. Bartels, N. Mauser; Institute of Analysis and Scientific Computing, 2022; oral examination: 2022-03-17.

Time-dependent micromagnetic phenomena are usually described by the nonlinear Landau-Lifshitz-Gilbert equation (LLG). The driving force of LLG is the effective field, which is derived from the micromagnetic energy functional, usually coupled to other partial differential equations like the Maxwell system.

This thesis addresses some of the numerical challenges of the reliable and efficient integration of LLG. The focus is on three families of finite element-based numerical schemes that are proven to be (unconditionally) convergent towards a weak solution of the problem: the midpoint scheme by Bartels & Prohl (2006), the tangent plane scheme by Alouges (2008), and two
recent predictor-corrector methods by Kim & Wilkening (2018). We extend the tangent plane scheme and the midpoint scheme to more general energy contributions, covering the non-standard Dzyaloshinskii-Moriya interaction (DMI) energy, which is the essential ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. Our constructive convergence analysis proves the existence of weak solutions in presence of DMI, which - to the best of our knowledge - was missing in the literature. Our numerical experiments hint that in most scenarios the cheaper tangent plane scheme provides sufficiently accurate simulations. In a comparative numerical study we observe, however, that for very sensitive dynamics crucially relying on an accurate energy evolution the midpoint scheme yields the most reliable results.

We propose and analyze strategies for the efficient solution of discrete systems obtained from the tangent plane scheme or from the midpoint scheme: We provide - to the best of our knowledge - the first rigorous analysis of the nonlinear midpoint scheme for three dimensional micromagnetics linearized by Newton's method. Further, for the constrained and non-symmetric linear system arising from the tangent plane scheme and posed in the time-dependent discrete tangent space, we derive possible preconditioning strategies, which guarantee linear convergence of the preconditioned GMRES algorithm.

For the predictor-corrector methods recently proposed by Kim & Wilkening, we close a fundamental gap in the original work by establishing unconditional well-posedness of the schemes. Our analysis even covers the case of a vanishing Gilbert damping parameter. Moreover, we propose implicit-explicit versions of the predictor-corrector methods, drastically reducing the computational cost, while preserving the formal convergence order.

All our theoretical contributions are accompanied by supportive numerical experiments. As a central element of the thesis, inspired by our theoretical findings, an easy-to-use open-source software module for the simulation of micromagnetic phenomena is developed: The module Commics is based on the multiphysics finite element library NGSolve, is freely available on GitLab, implements the state-of-the-art finite element integrators discussed in this thesis, and thus provides a means to enhance and promote research on the numerical integration of LLG.

https://www.asc.tuwien.ac.at/praetorius/download/thesis/phd/pfeiler2022.pdf