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

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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.