An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

Diagramas de fase J1(τ) y J1(γ) de un filme superconductor

En esta contribución utilizamos la teoría de Ginzburg-Landau dependiente del tiempo en presencia de corrientes a campo magnético nulo para investigar la dinámica de vórtices cinemáticos en una lámina superconductoras mesoscópicas con un pilar central delgado. Nuestro estudio abarca dos casos: (a) un pilar central al cual variamos su altura, simulada mediante el parámetro T > 1;0, con una interfase superconductor-vacío en toda la muestra, simulada con el parámetro γ = 1;0; (b) un pilar central con una condición de contorno superconductor-superconductor a mayor temperatura crítica Tc, (γ > 1;0); consideramos también una muestra homogénea, es decir sin pilar T= 1;0. Analizamos la influencia de diferentes condiciones de contorno en el estado de vórtice cinemáticos y sus efectos en la respuesta magnética mediante el análisis de las curva corriente-voltaje y resistividad-corriente; también es calculada la velocidad de aniquilación de los pares vórtice-anti vórtice en función de la corriente aplicada para varias condiciones de contorno. Los resultados muestran que las corrientes críticas y la dinámica de la aniquilación de vórtices cinemáticos son altamente dependiente de la altura del pilar y de las condiciones de contorno

Error bounds for discrete minimizers of the {G}inzburg--{L}andau energy in the high-κ\kappa regime

In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter $\kappa$. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ and the spatial mesh width $h$. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted $\kappa$-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived $L^2$- and $H^1$-error estimates are indeed optimal in $\kappa$ and $h$

A hierarchy of models for type-II superconductors

A hierarchy of models for type-II superconductors is presented. Through appropriate asymptotic limits we pass from the mesoscopic Ginzburg-Landau model to the London model with isolated superconducting vortices as line singularities, to vortex-density models, and finally to macroscopic critical-state models

Error bounds for discrete minimizers of the Ginzburg-Landau energy in the high-κ\kappa regime

In this work, we study discrete minimizers of the Ginzburg-Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg-Landau parameter $\kappa$. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ and the spatial mesh width $h$. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted $\kappa$-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we further explore the asymptotic optimality of our derived $L^2$- and $H^1$-error estimates with respect to $\kappa$ and $h$. Preasymptotic effects are observed for large mesh sizes $h$

Preconditioned Recycling Krylov subspace methods for self-adjoint problems

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used