Adaptive multiscale methods for the Ginzburg–Landau equations of superconductivity

该项目的目标是发展数值多尺度方法，以有效地求解超导的金兹堡-朗道方程。超导体是在足够低的温度下电阻消失的材料。在物理和工程上有许多应用，例如，超导磁体可以产生粒子加速器所需的极强磁场。特别令人感兴趣的是所谓的ii型超导体，它允许超导材料部分被磁场穿透。磁场穿透材料的孤立点可以形成复杂的涡旋晶格，具有非常精细的微观结构。相应的模拟，允许准确地预测或再现超导体的行为往往是复杂的，具有很高的计算复杂性。这有几个原因。一方面，底层的金兹堡-朗道方程通常是时变的、耦合的和非线性的，这自然使计算变得复杂。由于需要非常精细的计算网格来解析所有微结构，因此在复杂涡晶格的存在下，这变得更加严重。如果超导材料具有复杂的几何结构，则会产生额外的复杂性，这降低了解的总体规律性，从而限制了数值近似的预期收敛速率。该项目旨在发展合适的数值多尺度方法，允许将特定问题的结构（如漩涡）直接集成到某些广义有限元空间（“多尺度空间”）中。基于该方法，可以在低维多尺度空间中有效求解金兹堡-朗道方程，克服了现有方法的局限性。除了验证所开发的方法和相应的先验误差分析外，我们还计划将这些方法与允许自适应误差控制的技术相结合。

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.