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The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. This equation exhibits highly nonlinear behavior and involves a non-convex manifold constraint, namely a point-wise unit length preservation, along with energy dissipation property. It is well known that the normalized tangent plane method is able to simultaneously achieve the non-convex manifold constraint and original energy dissipation. However, the associated computational cost of this numerical approach is exceedingly high. By contrast, the projection method is more straightforward to implement, while it often compromises the inherent energy dissipative property of the continuous model, and the error analysis turns out to be even more challenging. In this work, we first construct a linear and fully discrete finite difference numerical scheme, based on the projection method for the LLG equation, which is capable of simultaneously preserving the point-wise unit length and an unconditional original energy dissipation. In the error analysis, the classical theoretical technique becomes ineffective, due to the presence of the nonlinear Laplacian term, which in turn poses a significant challenge. To overcome this subtle difficulty, we carefully rewrite the numerical method in an equivalent weak form, in which a point-wise length preserving feature of the numerical solution plays an essential role. Based on such a reformulation, a nonlinear Laplacian estimate is avoided, and the rest nonlinear error bounds could be derived with the help of discrete Sobolev interpolation, as well as a Law-of-Cosine style estimate of the numerical errors at the renormalization stage. As a result of these estimates in the reformulated weak form, an optimal convergence rate could be theoretically established. In our knowledge, this numerical method is the first linear algorithm that preserves the following combined theoretical properties: (i) point-wise length preservation, (ii) unconditional original energy dissipation, (iii) a theoretical justification of convergence analysis and optimal rate error estimate. Some numerical experiments are presented to verify the theoretical findings and illustrate the robustness and effectiveness of the proposed method.

Dr. Cheng Wang is a professor in Department of Mathematics at the University of Massachusetts Dartmouth (UMassD). He obtained hid Ph.D degree from Temple University in 2000, under the supervision of Prof. Jian-Guo Liu. Prior to joining UMassD in 2008 as an assistant professor, he was a Zorn postdoc at Indiana University from 2000 to 2003, under the supervision of Roger Temam and Shouhong Wang, and he worked as an assistant professor at University of Tennessee at Knoxville from 2003 to 2008. Dr. Wang’s research interests include development of stable, accurate numerical algorithms for partial differential equations and numerical analysis. He has published more than 140 papers with more than 9000 citations. He also serves in the Editorial Board of “Numerical Mathematics: Theory, Methods and Applications”.