Grid generation and multigrid schemes for spectral element methods in curved domains

该项目的目标是使谱元方法在涉及曲线域的应用中与经典的低阶离散化方法相竞争。为了实现这一目标，我们提出了一种综合方法，解决了以前方法的主要弱点。关键部分如下：1)从曲面的线性三角剖分出发，构建近等距斑块的稳定收敛方法。2)一种内置度量控制的生成符合边界曲线网格的移动网格方法。离散化运动网格和不可压缩流动问题中隐式方程的近最优复杂度谱元多重网格技术。1)中的一个主要挑战是设计如何构建表面顶点之间的边界曲线，以便在保持最佳精度的同时帮助创建行为良好的补丁。在此前提下，通过对采样点进行适当的参数化，优化patch构建本身。这种方法可以在不损失精度的情况下改进度量。一旦表面网格固定，移动网格法就可以生成一个性能良好的、边界拟合的空间网格。与Persson和Preraire最近提倡的弹性类比相比，运动网格法由于材料单元之间的松散耦合，具有直接度量控制和更高灵活性的优点。将解决的一个开放问题是适当地施加随时间变化的边界条件，它驱动从多面体向弯曲网格的过渡。此外，我们的目标是有效的隐式求解技术的谱元公式的移动网格方法。在该项目的第三部分中，结构开发多网格方法的发展将大大推动这项工作。最后，这些方法将被用作谱元流求解器原型的构建块，以演示整个方法的能力。

The objective of the project is to render the spectral element method competitive to classical lower order discretization methods for applications involving curved domains. To achieve this, we propose an integral approach that tackles the principal weaknesses of previous methods. The key components are as follows: 1) Stable and convergent methods for constructing nearly isometric patches starting from a linear triangulation of the surface.2) A moving mesh method for generating a boundary-conforming curved grid with built-in metric control.3) Spectral element multigrid techniques with near optimal complexity for implicit equations arising from discretization moving mesh and incompressible flow problems. A major challenge in 1) is to devise how boundary curves between surface vertices must be constructed in order to aid the creation of well-behaved patches while maintaining optimal accuracy. Based on this prerequisite, the patch construction itself will be optimized via suitable parameterization of sampling points. This approach allows for improving the metric without loss of precision. Once the surface grid is fixed, the moving mesh method serves to generate a well-behaved, boundary fitting spatial grid. In contrast to the elastic analogy recently advocated by Persson and Preraire, the moving mesh method has the advantage of direct metric control and higher flexibility because of the loose coupling between material elements. An open issue that will be addressed is the proper imposition of time-dependent boundary conditions, which drive the transition from the polyhedral toward a curved grid. In addition to this we aim at efficient implicit solution techniques for the spectral element formulation of the moving mesh method. This work will be fueled considerably by the development of structure-exploiting multigrid methods in the third part of the project. Finally these methods will be used as a building block for prototyping a spectral element flow solver that demonstrates the capability of the overall approach.