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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation


Authors: Eugenia Kim and Jon Wilkening
Journal: Quart. Appl. Math. 76 (2018), 383-405
MSC (2010): Primary 65M60, 35Q60; Secondary 78M10
DOI: https://doi.org/10.1090/qam/1485
Published electronically: September 21, 2017
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Abstract: The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.


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Additional Information

Eugenia Kim
Affiliation: Department of Mathematics, University of California, Berkeley, CA 94720
Email: kim107@math.berkeley.edu

Jon Wilkening
Affiliation: Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
Email: wilkening@berkeley.edu

DOI: https://doi.org/10.1090/qam/1485
Keywords: Landau-Lifshitz equation, Landau-Lifshitz-Gilbert equation, micromagnetics, finite element methods, mass-lumped method, convergence, weak solutions
Received by editor(s): August 8, 2017
Published electronically: September 21, 2017
Additional Notes: The first author was supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-AC05-06OR23100.
The second author was supported in part by the U.S. Department of Energy, Office of Science, Applied Scientific Computing Research, under award number DE-AC02-05CH11231.
Article copyright: © Copyright 2017 Brown University

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