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Mathematical and numerical analysis of the time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition

Authors: Buyang Li and Zhimin Zhang
Journal: Math. Comp. 86 (2017), 1579-1608
MSC (2010): Primary 65M12, 65M60; Secondary 35Q56, 35K61
Published electronically: November 18, 2016
MathSciNet review: 3626529
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Abstract: We prove well-posedness of the time-dependent Ginzburg–Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in $H^1(\Omega )$ in general, and so the finite element method (FEM) may give incorrect solutions. To overcome this difficulty, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit $H^1$ solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on the proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.

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Additional Information

Buyang Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
MR Author ID: 910552

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China — and — Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Superconductivity, reentrant corner, singularity, well-posedness, finite element method, convergence, Hodge decomposition
Received by editor(s): April 17, 2015
Received by editor(s) in revised form: February 8, 2016
Published electronically: November 18, 2016
Additional Notes: The work of the first author was supported in part by National Natural Science Foundation of China (NSFC) under grant 11301262. The research stay of this author at Universität Tübingen was funded by the Alexander von Humboldt Foundation
The second author is the corresponding author, whose work was supported in part by the National Natural Science Foundation of China (NSFC) under grants 11471031, 91430216, and U1530401, and by the US National Science Foundation (NSF) through grant DMS-1419040.
Article copyright: © Copyright 2016 American Mathematical Society