Multiscale modeling in micromagnetics: existence of solutions and numerical integration

Abstract

Various applications ranging from spintronic devices, giant magnetoresistance sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau-Lifshitz-Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but cannot describe the short-range exchange interaction accurately. A combination of both methods allows to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a non-linear material law using the Maxwell equations. In our work, we prove that under certain assumptions on the non-linear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak $H^1$-convergence (at least for a subsequence) towards a weak solution.

35 pages

References in corpus (3)

• A convergent and precise finite element scheme for Landau-Lifschitz-Gilbert equation
• Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity
• Each H^{1/2}-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in R^d

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• Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation
• Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics
• Convergence of an implicit-explicit midpoint scheme for computational micromagnetics
• Multiscale Model Approach for Magnetization Dynamics Simulations
• Computational micromagnetics with Commics
• Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics
• Optimal convergence rates of an adaptive hybrid FEM-BEM method for full-space linear transmission problems