Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg--Landau equations of superconductivity

TL;DR

This paper analyzes a linearized Galerkin-mixed finite element method for time-dependent Ginzburg--Landau equations, demonstrating optimal error estimates and improved convergence on complex domains compared to traditional methods.

Contribution

It introduces a mixed FE scheme with an optimal error estimate that converges unconditionally under broader regularity assumptions, including nonconvex and polyhedral domains.

Findings

The mixed method achieves optimal error bounds.

Numerical examples confirm efficiency on complex domains.

The scheme converges unconditionally under general regularity conditions.

Abstract

A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg--Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field ${\sigma} = \mathrm{curl} \, {\bf{A}}$ as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional noncovex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains.