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

Buyang Li, Zhimin Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)

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 H1(Ω) 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 H1solutions 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.
Original languageEnglish
Pages (from-to)1579-1608
Number of pages30
JournalMathematics of Computation
Volume86
Issue number306
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • Convergence
  • Finite element method
  • Hodge decomposition
  • Reentrant corner
  • Singularity
  • Superconductivity
  • Well-posedness

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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