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

TL;DR

This paper establishes well-posedness for the time-dependent Ginzburg--Landau equations in nonconvex polygons, reformulates the problem using Hodge decomposition to enable accurate finite element solutions, and validates the approach with numerical experiments.

Contribution

It introduces a reformulation of the Ginzburg--Landau equations using Hodge decomposition to handle nonconvex domains and magnetic potential regularity issues, enabling correct finite element solutions.

Findings

Reformulated equations admit $H^1$ solutions for key variables.

Proposed a decoupled, linearized FEM with proven error estimates.

Numerical examples confirm theoretical accuracy and efficiency.

Abstract

We prove well-posedness of 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 $H^1$ in general, and the finite element method (FEM) may give incorrect solutions. To remedy this situation, 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 proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.