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Analysis and numerical methods for multiscale problems in magnetization dynamics

Time: Fri 2021-12-10 15.00

Location: Sal F3 och , Lindstedtsvägen 26

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Language: English

Subject area: Applied and Computational Mathematics, Numerical Analysis

Doctoral student: Lena Leitenmaier , Numerisk analys, NA

Opponent: Professor Carlos Garcia-Cervera,

Supervisor: Professor Olof Runborg, Numerisk analys, NA

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This thesis investigates a multiscale version of the Landau-Lifshitz equation and how to solve it using the framework of Heterogeneous Multiscale Methods (HMM). The Landau-Lifshitz equation is the governing equation in micromagnetics, modeling magnetization dynamics. The considered problem involves two different scales which interact with each  other: a fine scale, on which small material variations can be described, and a coarse scale for the overall magnet. Since the fast variations are much smaller than the coarse scale, the computational cost of resolving  these scales in a direct numerical simulation is very high. The idea behind HMM therefore is to use a coarse macro model,  involving some missing quantity, in combination with an exact micro model that provides the information necessary to complete the macro model using an averaging process, the so-called  upscaling. This approach results in a computational cost that is independent of the fine scale, ε.

The included papers focus on different aspects of the problem, together providing both error estimates and implementation details.

Paper I investigates homogenization of the given Landau-Lifshitz  equation with a rapidly oscillating material coefficient in a periodic setting. Equations for the homogenized solution and the corresponding correctors are derived and estimates for the error introduced by homogenization are given. Both the difference between actual and homogenized solution as well as corrected approximations are considered. We show convergence rates in ε up to final times  T ∈ O(εσ), where  0 <  σ ≤ 2, in Hq Sobolev norms. Here the choice of  q is only restricted by the regularity of the solutions.

In Paper II, three different ways to set up HMM are introduced, the  flux, field and torque model. Each model involves a different  missing quantity in the HMM macro model.  In a periodic setting,  the errors introduced when approximating the missing  quantities are analyzed. In all three models  the upscaling errors are bounded similarly and can be reduced to O(ε) when choosing the involved parameters  optimally.

A finite difference based implementation of the field model is studied in Paper III. Several important aspects, such as choice of  time integrator, size of the micro domain, boundary conditions for the micro problem and the influence of various parameters introduced in the upscaling process are discussed. We moreover introduce the idea to use artificial damping in the micro problem to obtain a more efficient implementation.

Finally, a more physical setup is considered in Paper IV. A finite element macro model that is combined with a finite difference  micro model is proposed. This approach is based on a variation of  the flux model introduced in Paper II. A problem setting with  Neumann boundary conditions and involving several terms in the so-called effective field is considered. Numerical examples show  the viability of the approach.

Additionally, several geometric time integrators for the Landau-Lifshitz equation are reviewed and compared in a technical  report. Their properties are investigated using numerical examples.