The mimetic finite difference method for the Landau–Lifshitz equation

Kim, Eugenia; Lipnikov, Konstantin

doi:10.1016/j.jcp.2016.10.016

Cited by 13 publications

(9 citation statements)

References 48 publications

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“…The method is also shown to be useful for meshes with degenerate and nonconvex polygonal elements. Since then, MFD method has been implemented in various problems (diffusion/convection-diffusion [101][102][103][104], electromagnetic field problems [105,106], elasticity problems, Lagrangian hydrodynamics problems [107], and solving wave equations [108]) and for modelling fluid flows [109,110]. MFD has been extended to higher order [100,102] and also to 3D [102,105,111,112].…”

Section: Mimetic Finite Difference (Mfd) Methods and Virtualmentioning

confidence: 99%

A Brief Review on Polygonal/Polyhedral Finite Element Methods

Perumal

2018

Mathematical Problems in Engineering

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This paper provides brief review on polygonal/polyhedral finite elements. Various techniques, together with their advantages and disadvantages, are listed. A comparison of various techniques with the recently proposed Virtual Node Polyhedral Element (VPHE) is also provided. This review would help the readers to understand the various techniques used in formation of polygonal/polyhedral finite elements.

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Section: Mimetic Finite Difference (Mfd) Methods and Virtualmentioning

confidence: 99%

A Brief Review on Polygonal/Polyhedral Finite Element Methods

Perumal

2018

Mathematical Problems in Engineering

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“…There has been a continuous progress of developing numerical algorithms in the past few decades; see for example [6,7] and references therein. The spatial derivative is typically approximated by the finite element method (FEM) [8,9,10,11,12,13,14,15,16,17,18] and the finite difference method [19,20,21,22,23]. As for the temporal discretization, explicit schemes [15,24], fully implicit schemes [25,26,20], and semi-implicit schemes [19,27,28,29,30,31,32] have been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

Second-order semi-implicit projection methods for micromagnetics simulations

Xie

Garcı́a-Cervera

Wang

et al. 2020

Journal of Computational Physics

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Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

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“…Again, a nonlinear solver is necessary at each time step, and the same step-size condition has to be imposed. The existing works of finite difference method to the LL equation may be referred to [18,19,23,26,44]. In [18], a time stepping method in the form of a projection method was proposed; this method is implicit and unconditionally stable, and the rigorous proof was provided with the first-order accuracy in time and second-order accuracy in space.…”

Section: Introductionmentioning

confidence: 99%

“…In [23], an updated source term was used, and an iteration algorithm was repeatedly performed until the numerical solution converges. In [26], the explicit and implicit mimetic finite difference algorithm was developed.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of A Second-order Semi-implicit Projection Method for Landau-Lifshitz Equation

Chen

Cheng

Xie

2019

Preprint

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The numerical approximation for the Landau-Lifshitz equation, the dynamics of magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by introducing two sets of approximated solutions to preceed estimation alternatively, with unconditional stability and second-order accuracy in both time and space, provided that the spatial step-size is the same order as the temporal step-size, which remarkably relax restrictions of temporal step-size compared to the implicit schemes. And also, the unique solvability of the numerical solution without any assumptions for the step size in both time and space is theoretically justified, which turns out to be the first such result for the micromagnetics model. All these theoretical properties are verified by numerical examples in both oneand three-dimensional spaces.

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The mimetic finite difference method for the Landau–Lifshitz equation

Cited by 13 publications

References 48 publications

A Brief Review on Polygonal/Polyhedral Finite Element Methods

A Brief Review on Polygonal/Polyhedral Finite Element Methods

Second-order semi-implicit projection methods for micromagnetics simulations

Convergence Analysis of A Second-order Semi-implicit Projection Method for Landau-Lifshitz Equation

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