Adaptive multiscale methods for the Ginzburg–Landau equations of superconductivity

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Goal of the project is the development of numerical multiscale methods for an efficient solving of the Ginzburg-Landau equations of superconductivity. Superconductors are materials with a vanishing electrical resistance at sufficiently low temperatures. There are numerous applications in physics and engineering, for example, superconducting magnets can generate extremely strong magnetic fields that are needed for particle accelerators. Of particular interest are so-called type-II superconductors which allow that the superconducting material is partially penetrated by magnetic fields. The isolated points in which the magnetic field penetrates the material can form a complex vortex lattice with a very fine microstructure. Corresponding simulations that allow to accurately predict or reproduce the behavior of superconductors are often complicated and have a high computational complexity. This has several reasons. On the one hand, the underlying Ginzburg-Landau equations are typically time-dependent, coupled and nonlinear, which naturally complicates the computations. This becomes more severe in the presence of a complex vortex lattice since it requires very fine computational meshes to resolve all microstructures. Additional complications can arise if the superconducting material has a complex geometry, which reduces the overall regularity of solutions and hence limits the expected convergence rates of numerical approximations. The project aims at the development of suitable numerical multiscale methods that allow to integrate problem-specific structures (such as vortices) directly into certain generalized finite element spaces ("multiscale spaces"). Based on this approach, the Ginzburg-Landau equations can be solved efficiently in these low dimensional multiscale spaces, helping to overcome the limitations of existing methods. Besides the validation of the developed approaches and a corresponding a priori error analysis, we also plan to combine the methods with techniques that allow for an adaptive error control.