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This article is concerned with the integration of the time-dependent Ginzburg– Landau (TDGL) equations of superconductivity. Four algorithms, ranging from fully explicit to fully implicit, are presented and evaluated for stability, accuracy, and compute time. The benchmark problem for the evaluation is the equilibration of a vortex configuration in...
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Consider a class of Sobolev functions satisfying a prescribed degree condition on the boundary of a planar annular domain. It is shown that the minimizers of the Ginzburg-Landau functional in this class are radially symmetric when the annulus is sufficiently narrow.
Citations
... We refer to [8,9] for theoretical analysis for smooth domains. Numerical methods have been studied extensively, see [6,7,10,11,13,14,16,19,20,[31][32][33]35]. It is well known that boundary defect has a significant influence on the superconductors [1,4]. ...
... Numerical analysis for TDGL system under the Lorentz gauge has been done extensively, see [6, 8, 9, 19, 20, 22, 23, 29-31, 33, 38], while the temporal gauge is favored by practitioners due to its lower computational cost, see [1,12,15,18,21,25,28,35,36,40,42,43]. For the TDGL equations under the Coulomb gauge, the well-posedness was studied in [26,41]. ...
This paper is concerned with numerical analysis of a finite element method for the time-dependent Ginzburg–Landau equations under the Coulomb gauge. The main challenge is that the magnetic potential \({{\varvec{A}}}\) is divergence-free and satisfies a Stokes-like structure under the Coulomb gauge. The proposed method uses linear Lagrange element \({\mathcal {P}}_1\) to solve for the order parameter \(\psi \), the lowest order Nédélec edge element \(\mathcal {N}\! \mathcal {D}_{\! 1}\) and linear Lagrange element \({P}_1\) to approximate the magnetic potential \({{\varvec{A}}}\) and the electric potential \(\phi \), respectively. In particular, the proposed method preserves a weakly divergence-free property for \({{\varvec{A}}}\) in the discrete level. The main aim of this work is to establish the second order spatial convergence of the most important variable \(\psi \), though the numerical solutions of \({{\varvec{A}}}\) are only O(h) in space. Our analysis is based on a nonstandard quasi-projection for \(\psi \) and the corresponding \(H^{-1}\)-norm estimates for Maxwell projection. With the quasi-projection, we prove that the lower-order approximation to \({{\varvec{A}}}\) does not pollute the accuracy of \(\psi _h\). An effective one step recovery is also proposed to obtain second order numerical solution for \({{\varvec{A}}}\). Our numerical experiments confirm the optimal second order convergence of \(\psi _h\) and the efficiency of the recovery step.
... [1]), and it has been widely applied in physics and engineering; there have been numerous literature available on the mathematical issues, the numerical methods and the applications, cf. [2][3][4][5][6][7][8][9][10][11], just to name a few. We are concerned with the finite element method for the numerical solution of the time-dependent GL (TDGL) model as follows: ...
We propose and analyze residual-based a posteriori error estimator for a new finite element method for the time-dependent Ginzburg–Landau equations with the temporal gauge of superconductivity. The magnetic potential variable is approximated by H(∇×;Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\nabla \times ;\Omega )$$\end{document}-conforming element and the scalar order parameter is approximated by the H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega )$$\end{document}-conforming element. Using the dual problem of a linearization of the original problem, we prove the reliability of the a posteriori error estimator, and an adaptive algorithm with the temporal and spatial refining and coarsening steps is then proposed. Numerical results are presented for illustrating the a posteriori error estimator and the adaptive algorithm in convex and nonconvex domains.
... Motivated by this, gauge-invariant schemes have been developed and widely used in practice, which can preserve the point-wise boundedness. One of the typical examples is the gauge-invariant finite difference method, see [3,9,11,15,[21][22][23][24]35] for its analysis and applications. We shall also mention that a co-volume approximation for the steady state GL equation had been studied in [14]. ...
In this paper, we propose and analyze a stabilized semi-implicit Euler gauge-invariant method for numerical solution of the time-dependent Ginzburg–Landau (TDGL) equations in the two-dimensional space. The proposed method uses the well-known gauge-invariant finite difference approximations with staggered variables in a rectangular mesh, and a stabilized semi-implicit Euler discretization for time integration. The resulted fully discrete system leads to two decoupled linear systems at each time step, thus can be efficiently solved. We prove that the proposed method unconditionally preserves the point-wise boundedness of the solution and is also energy-stable. Moreover, the proposed method under the zero-electric potential gauge is shown to be equivalent to a mass-lumped version of the lowest order rectangular Nédélec edge element approximation and the Lorentz gauge scheme to a mass-lumped mixed finite element method. These indicate the method is also effective in solving the TDGL problems in non-convex domains although the solutions are often of low-regularity in such situation. Various numerical experiments are also presented to demonstrate effectiveness and robustness of the proposed method.
... To solve the TDGL equation, the explicit Euler method is used. In this case, we consider the maximum allowable timestep size ∆t that stabilizes the iteration [49]. We find that ∆t decreases with (i) a smaller grid spacing, (ii) the inclusion of ψ gra , and (iii) larger gradient energy coefficients η αβ g0 . ...
... The global existence and uniqueness of the strong solution were established in [13] for the TDGL equations with the Lorentz gauge. Numerical methods for solving the TDGL equations have also been studied extensively, e.g., see [2,12,19,21,25,28,32,33,41,42,45,49]. A semi-implicit Euler scheme with a finite element approximation was first proposed by Chen and Hoffmann [12] for the TDGL equations with the Lorentz gauge. ...
... If the superconductor is a long cylinder in the z-direction with a finite cross section and the external applied field H e = H e [0, 0, 1] T (i.e., H e is parallel to the z-axis), the original three dimensional equations (1.1)-(1.2) can be reduced to a two dimensional equation [33,45] ∂ψ ...
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.
... Numerical analysis of the TDGL under the zero electric potential gauge φ = 0 has also been done in many works [5,26,28,33,[39][40][41]43]; also see the review paper [20]. Since ∇ × A L 2 is not equivalent to ∇A L 2 , both theoretical and numerical analysis are difficult under this gauge without extra assumptions on the regularity of the PDE's solution. ...
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.
... In most computational schemes used for TDGL, the system is discretized in space on a regular grid and integrated in time by using an explicit or implicit time integration scheme (see e.g. Gropp et al (1995), Crabtree et al (2000 and Gunter et al (2002)). A recent implementation using an underlying regular spatial mesh on modern graphics processing units is described in detail in Sadovskyy et al (2015). ...
The behavior of vortex matter in high-temperature superconductors (HTS) controls the entire electromagnetic response of the material, including its current carrying capacity. Here, we review the basic concepts of vortex pinning and its application to a complex mixed pinning landscape to enhance the critical current and to reduce its anisotropy. We focus on recent scientific advances that have resulted in large enhancements of the in-field critical current in state-of-the-art second generation (2G) YBCO coated conductors and on the prospect of an isotropic, high-critical current superconductor in the iron-based superconductors. Lastly, we discuss an emerging new paradigm of critical current by design-a drive to achieve a quantitative correlation between the observed critical current density and mesoscale mixed pinning landscapes by using realistic input parameters in an innovative and powerful large-scale time dependent Ginzburg-Landau approach to simulating vortex dynamics.
... However, if the material is a long cylinder in the z-direction with a finite cross section and the external applied magnetic field H e = H e [0, 0, 1] T (i.e., H e is parallel to the z-axis), the original three-dimensional equations (2.2)-(2.5) can be reduced to a two-dimensional system [27,42,43], see Fig. 1 for illustration. ...
... Since in this case, the three-dimensional TDGL equations can be simplified into the two-dimensional TDGL equations (2.6)-(2.9), see [27,42,43], numerical results presented here are similar to the results obtained in Example 4.2 due to the same data setting. ...
The paper focuses on numerical study of the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge. The proposed method is based on a fully linearized backward Euler scheme in temporal direction, and a mixed finite element method (FEM) in spatial direction, where the magnetic field is introduced as a new variable. The linearized Galerkin-mixed FEM enjoys many advantages over existing methods. In particular, at each time step the scheme only requires the solution of two linear systems for ψ and , respectively, with constant coefficient matrices. These two matrices can be pre-assembled at the initial time step and these two linear systems can be solved simultaneously. Moreover, the method provides the same order of optimal accuracy for the density function ψ, the magnetic potential A, the magnetic field , the electric potential and the current . Extensive numerical experiments in both two- and three-dimensional spaces, including complex geometries with defects, are presented to illustrate the accuracy and stability of the scheme. Our numerical results also show that the proposed method provides more realistic predictions for the vortex dynamics of the TDGL equations in nonsmooth domains, while the vortex motion influenced by a defect of the domain is of high interest in the study of superconductors.
... and convergence of the numerical solutions have been reviewed in [13] and an alternating Crank-Nicolson schemes was proposed in [23]. Some implicit, explicit and implicit-explicit time discretization schemes were studied in [19]. For the finite element approximations, error estimates were carried out for a regularized problem, by adding a term − ∇(∇ · A) to the equation (1.8). ...
We introduce a new approach for finite element simulations of the
time-dependent Ginzburg-Landau equations (TDGL) in a general curved polygon,
possibly with reentrant corners. Specifically, we reformulate the TDGL into an
equivalent system of equations by decomposing the magnetic potential to the sum
of its divergence-free and curl-free parts, respectively. Numerical simulations
of vortex dynamics show that, in a domain with reentrant corners, the new
approach is much more stable and accurate than the old approaches of solving
the TDGL directly (under either the temporal gauge or the Lorentz gauge); in a
convex domain, the new approach gives comparably accurate solutions as the old
approaches.
