Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/74457
Title: Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data
Authors: Li, B 
Keywords: Convergence
Finite element method
Ginzburg–Landau
Incompatible data
Nonconvex polyhedra
Superconductivity
Issue Date: 2017
Publisher: Springer
Source: Calcolo, 2017, v. 54, no. 4, p. 1441-1480 How to cite?
Journal: Calcolo 
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 (Formula presented.) regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a (Formula presented.) 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.
URI: http://hdl.handle.net/10397/74457
ISSN: 0008-0624
EISSN: 1126-5434
DOI: 10.1007/s10092-017-0237-0
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