@Article{CiCP-28-2,
author = {Li, Buyang and Wang, Kai and Zhimin, Zhang},
title = {A Hodge Decomposition Method for Dynamic Ginzburg–Landau Equations in Nonsmooth Domains — A Second Approach},
journal = {Communications in Computational Physics},
year = {2020},
volume = {28},
number = {2},
pages = {768--802},
abstract = {<p style="text-align: justify;">In a general polygonal domain, possibly nonconvex and multi-connected
(with holes), the time-dependent Ginzburg–Landau equation is reformulated into a
new system of equations. The magnetic field $B$:=∇×<strong>A</strong> is introduced as an unknown
solution in the new system, while the magnetic potential <strong>A</strong> is solved implicitly through
its Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method
is proposed for solving the new system. The convergence of numerical solutions is
proved based on a compactness argument by utilizing the maximal $L^p$-regularity of
the discretized equations. Compared with the Hodge decomposition method proposed in [27]<sub>,&nbsp;</sub>the new method has the advantage of approximating the magnetic field
B directly and converging for initial conditions that are incompatible with the external
magnetic field. Several numerical examples are provided to illustrate the efficiency of
the proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new method
is superior to the method in [27] for approximating the magnetic field.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2019-0117},
url = {https://global-sci.com/article/79703/a-hodge-decomposition-method-for-dynamic-ginzburglandau-equations-in-nonsmooth-domains-a-second-approach}
}