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

TL;DR

This paper introduces an efficient linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau equations, demonstrating optimal complexity and independence from problem dimension through numerical experiments.

Contribution

It develops a preconditioned Newton--Krylov method leveraging properties of the Jacobian and AMG strategies, achieving optimal solver complexity for this nonlinear PDE.

Findings

Krylov iteration count is independent of solution dimension

Solver complexity is O(n) for discretized problems

Numerical experiments confirm efficiency in 2D and 3D domains

Abstract

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).