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

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

Linear second-order IMEX-type integrator for the (eddy current) Landau–Lifshitz–Gilbert equation

Combining ideas from Alouges et al. (2014, A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation. Numer. Math., 128, 407–430) and Praetorius et al. (2018, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics. Comput. Math. Appl., 75, 1719–1738) we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau–Lifshitz–Gilbert (LLG) equation, which is unconditionally convergent, formally (almost) second-order in time, and requires the solution of only one linear system per time step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then we extend the scheme to the coupled system of the LLG equation with the eddy current approximation of Maxwell equations. Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires the solution of only two linear systems per time step

Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics

We consider the numerical approximation of the Landau–Lifshitz–Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii–Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes

Computational micromagnetics with Commics

We present our open-source Python module Commics for the study of the magnetization dynamics in ferromagnetic materials via micromagnetic simulations. It implements state-of-the-art unconditionally convergent finite element methods for the numerical integration of the Landau–Lifshitz–Gilbert equation. The implementation is based on the multiphysics finite element software Netgen/NGSolve. The simulation scripts are written in Python, which leads to very readable code and direct access to extensive post-processing. Together with documentation and example scripts, the code is freely available on GitLab

Convergence of an implicit–explicit midpoint scheme for computational micromagnetics

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau–Lifschitz–Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams–Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings

Numerische Integratoren zweiter Ordnung für die Landau-Lifshitz-Gilbert-Gleichung

Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersDie Landau-Lifshitz-Gilbert Gleichung (LLG) ist das fundamentale mathematische Modell für Verständnis und Simulation zeitabhängiger mikromagnetischer Phänomene. Schwierigkeiten bei der Entwicklung effizienter numerischer Verfahren sind die Nichtlinearität der Gleichung, eine nicht-konvexe Nebenbedingung, und die Nicht-Eindeutigkeit von Lösungen. Mit dem (zweite Ordnung) Tangent-Plane-Verfahren aus [Alouges et al. (Numer. Math., 128, 2014)] und dem Midpoint-Verfahren aus [Bartels and Prohl (2006) (SIAM J. Numer. Anal., 44)] verfügen wir über zwei Zeitschrittverfahren mit (formal) zweiter Konvergenzordnung in der Zeit. Beide Algorithmen basieren auf der Finite-Elemente-Methode und konvergieren unbedingt. Die spezielle Struktur beider Algorithmen legt bei Erweiterungen die aufwändige implizite Behandlung von etwaigen Termen niedriger Ordnung und von gekoppelten Gleichungen nahe, beispielsweise Streufeld-Berechnungen oder die Kopplung von LLG mit der Maxwell-Gleichung. Um dieses Problem zu umgehen, bedienen wir uns eines implizit-expliziten Adams-Bashforth-artigen Ansatzes, mit dem wir die Terme niedriger Ordnung explizit behandeln. Bei Kopplungen von LLG mit anderen Gleichungen entkoppeln wir die näherungsweise Berechnung der Magnetisierung (als Lösung von LLG) und der Lösung der gekoppelten Gleichung (z.B. elektrisches und magnetisches Feld bei der Kopplung von LLG mit der Maxwell-Gleichung). Die so erhaltenen Algorithmen sind (formal) von zweiter Ordnung in der Zeit. Für die Kopplung mit der Eddy-Current-Gleichung erhalten wir so ein entkoppeltes Tangent-Plane-Verfahren mit Konvergenz zweiter Ordnung in der Zeit. Für die Kopplung mit der Spin-Diffusion-Gleichung erhalten wir so ein entkoppeltes Midpoint-Verfahren mit Konvergenz zweiter Ordnung in der Zeit. Darüber hinaus organisieren wir die Annahmen beider Verfahren in einem einheitlichen Rahmen, der insbesondere physikalisch relevante dissipative Effekte abdeckt. Wir erweitern die bekannte numerische Analysis und beweisen die unbedingte Konvergenz all unserer erweiterten Algorithmen. Zusätzlich behandeln wir Lösungsstrategien für die entsprechenden Variationsformulierungen. Schließlich führen wir mit unseren erweiterten Algorithmen numerische Experimente durch. Diese Experimente bestätigen die Konvergenz zweiter Ordnung in der Zeit, den reduzierten Aufwand und die Anwendbarkeit auf physikalisch relevante Beispiele.In computational micromagnetism, the Landau-Lifshitz-Gilbert equation (LLG) is the fundamental mathematical model for the understanding and simulation of time-dependent micromagnetic phenomena. The non-linear nature of the equation, a non-convex side constraint, and the non-uniqueness of solutions aggravate the development of efficient numerical algorithms. The (second-order) tangent plane scheme from [Alouges et al. (Numer. Math., 128, 2014)] and the midpoint scheme from [Bartels and Prohl (2006) (SIAM J. Numer. Anal., 44)] provide us with two finite-element-based algorithms, which are both (formally) second-order in time and unconditionally convergent. The particular structure of both algorithms suggests the numerically expensive implicit treatment of possible lower-order terms and of coupled systems like, e.g., the computation of the stray field or, more general, the coupling of LLG with the full Maxwell system. To avoid this and to conserve the second-order in time convergence, we employ an implicit-explicit second-order in time Adams-Bashforth-type approach, where we treat the lower-order terms explicitly in time. For couplings with other equations, this decouples the approximate computation of the magnetization (i.e., the solution of LLG), and of the coupled equation (e.g., electrical and magnetic field of the coupling of LLG with the full Maxwell system). The resulting algorithms are formally second-order in time. For the coupling with eddy currents, this yields a decoupled second-order in time tangent plane scheme. For the coupling with the spin diffusion equation, this yields a decoupled second-order in time midpoint scheme. Moreover, we provide certain assumptions in a unified framework, which covers, in particular, physically relevant dissipative effects. We extend the existing convergence analysis and prove unconditional convergence of our extended algorithms. Moreover, we discuss the efficient solution of the corresponding (linear and non-linear) variational problems. Numerical experiments with our extensions confirm the preservation of the second-order in time convergence, reduced computational costs and the applicability to physically relevant examples of our algorithms.20