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

Li, Buyang; Zhang, Zhimin

doi:10.1090/mcom/3177

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