Convergence of a decoupled mixed FEM for the dynamic Ginzburg--Landau equations in nonsmooth domains with incompatible initial data

TL;DR

This paper develops a fully discrete mixed finite element method for the time-dependent Ginzburg--Landau equations in complex, nonsmooth domains, proving convergence without regularity assumptions and demonstrating superior performance over standard methods.

Contribution

It introduces a decoupled time-stepping scheme with convergence proof in complex domains, extending finite element analysis for Ginzburg--Landau equations.

Findings

Finite element solutions converge in nonsmooth, multi-connected domains.

The method guarantees bounded discrete energy and convergence.

Numerical results show improved convergence over standard Galerkin methods.

Abstract

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg--Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $L^{3+\delta}$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the N\'ed\'elec finite element space, and introducing a $\ell^2(W^{1,3+\delta})$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.