Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)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.
We propose and analyze a decoupled time-marching scheme for the coupling of the Landau–Lifshitz–Gilbert equation with a quasilinear diffusion equation for the spin accumulation. This model describes the interplay of magnetization and electron spin accumulation in magnetic and nonmagnetic multilayer structures. Despite the strong nonlinearity of the overall PDE system, the proposed integrator requires only the solution of two linear systems per time-step. Unconditional convergence of the integrator towards weak solutions is proved.
We solve a time-dependent three-dimensional spin-diffusion model coupled to the Landau-Lifshitz-Gilbert equation numerically. The presented model is validated by comparison to two established spin-torque models: The model of Slonzewski that describes spin-torque in multi-layer structures in the presence of a fixed layer and the model of Zhang and Li that describes current driven domain-wall motion. It is shown that both models are incorporated by the spin-diffusion description, i.e., the nonlocal effects of the Slonzewski model are captured as well as the spin-accumulation due to magnetization gradients as described by the model of Zhang and Li. Moreover, the presented method is able to resolve the time dependency of the spin-accumulation.
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.Date: November 26, 2018. 2010 Mathematics Subject Classification. 35R60, 65C20, 65N12, 65N15, 65N30. Key words and phrases. adaptive methods, a posteriori error analysis, convergence, two-level error estimate, stochastic Galerkin methods, finite element methods, parametric PDEs.Acknowledgements.
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).Adaptive techniques based on rigorous a posteriori error analysis of computed solutions provide an effective mechanism for building approximation spaces and accelerating convergence of computed solutions. These techniques rely heavily on how the approximation error is estimated and controlled. One may choose to estimate the error in the global energy norm and use the associated error indicators to enhance the computed solution and drive the energy error estimate to zero. However, in practical applications, simulations often target a specific (e.g., localized) feature of the solution, called the quantity of interest and represented using a linear functional of the solution. In these cases, the energy norm may give very little useful information about the simulation error.Alternative error estimation techniques, such as goal-oriented error estimations, e.g., by the dual-weighted residual methods, allow to control the errors in the quantity of interest. While for deterministic PDEs, these error estimation techniques and the associated adaptive algorithms are very well studied (see, e.g., [EEHJ95, JS95, BR96, PO99, BR01, GS02, BR03] for the a posteriori error estimation and [MS09, BET11, HP16, FPZ16] for a rigorous convergence analysis of adaptive algorithms), relatively little work has been done for PDEs with parametric or uncertain inputs. For example, in the framework of (intrusive) stochastic Galerkin finite element methods (sGFEMs) (see, e.g., [GS91, LPS14]), a posteriori error estimation of linear functionals of solutions to PDEs with parametric uncertainty is addressed in [MLM07] and, for nonlinear problems, in [BDW11]. In particular, in [MLM07], a rigorous estimator for the error in the quantity of interest is derived and several adaptive refinement strategies are discussed. However, the authors comment that the proposed estimator lacks information about the structure of the estim...
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) (Convergence of an implicit finite element method for the Landau-Lifschitz-Gilbert equation. SIAM J. Numer. Anal. 44(4):1405-1419), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lowerorder 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.
We propose a three-dimensional micromagnetic model that dynamically solves the Landau-Lifshitz-Gilbert equation coupled to the full spin-diffusion equation. In contrast to previous methods, we solve for the magnetization dynamics and the electric potential in a self-consistent fashion. This treatment allows for an accurate description of magnetization dependent resistance changes. Moreover, the presented algorithm describes both spin accumulation due to smooth magnetization transitions and due to material interfaces as in multilayer structures. The model and its finite-element implementation are validated by current driven motion of a magnetic vortex structure. In a second experiment, the resistivity of a magnetic multilayer structure in dependence of the tilting angle of the magnetization in the different layers is investigated. Both examples show good agreement with reference simulations and experiments respectively.
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