Optimal Analysis of Non-Uniform Galerkin-Mixed Finite Element Approximations to the Ginzburg–Landau Equations in Superconductivity

.This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg–Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart–Thomas face element are used for the order parameter \(\psi\) , the magnetic field \(\mathbf{curl}{\boldsymbol{A}}\) , and the magnetic potential \(\boldsymbol{A}\) , respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg–Landau model, the order parameter \(\psi\) is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of \((\mathbf{curl}{\boldsymbol{A}},{\boldsymbol{A}})\) influences the accuracy of \(\psi_h\) . The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for \((\mathbf{curl}{\boldsymbol{A}},{\boldsymbol{A}})\) is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for \(\psi\) and the corresponding more precise estimates, including the \(H^{-1}\) -norm. With the quasi-projection, we prove that the lower-order approximation to \((\mathbf{curl}{\boldsymbol{A}},{\boldsymbol{A}})\) does not pollute the accuracy of \(\psi_h\) . Our numerical experiments confirm the optimal convergence of \(\psi_h\) . The approach can be extended to many other multiphysics models.KeywordsGinzburg–Landau equationlowest–order approximationmixed finite elementoptimal error estimatesuperconductivityMSC codes65M60