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D. Praetorius, M. Ruggeri, B. Stiftner:
"Convergence of an implicit-explicit midpoint scheme for computational micromagnetics";
Computers and Mathematics with Applications, 75 (2018), 5; S. 1719 - 1738.

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 equires 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 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.

http://dx.doi.org/10.1016/j.camwa.2017.11.028