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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Mathematical and numerical analysis of the time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition
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by Buyang Li and Zhimin Zhang PDF
Math. Comp. 86 (2017), 1579-1608 Request permission

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
  • Email: buyang.li@polyu.edu.hk, libuyang@gmail.com
  • 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
  • Email: zmzhang@csrc.ac.cn, zzhang@math.wayne.edu
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1579-1608
  • MSC (2010): Primary 65M12, 65M60; Secondary 35Q56, 35K61
  • DOI: https://doi.org/10.1090/mcom/3177
  • MathSciNet review: 3626529