Analysis of Galerkin FEMs for Mixed Formulation of Time-Dependent Ginzburg--Landau Equations Under Temporal Gauge

Abstract

The paper focuses on analysis of linearized Galerkin FEMs for a mixed formulation of the time-dependent Ginzburg--Landau equations under the temporal gauge. We provide optimal error estimates in $L^2$-norm for the order parameter $\psi_h$ and the magnetic field $\sigma_h$ unconditionally, although the accuracy of the numerical magnetic potential $\mathbf{A}_h$ is one-order lower than the optimal one due to the degeneracy of the magnetic potential equation. Since the states of superconductors are determined by the order parameter $\psi_h$ (or the density of the superconducting electron pairs $|\psi_h|$), the accuracy of $\psi_h$ is more important for the vortex simulation in superconditors. Our analysis is based on a nonclassical Ritz projection, which may reduce the pollution of inaccuracy of the numerical magnetic potential in analysis. Numerical experiments confirm our theoretical analysis.