Legendre-Gauss-Lobatto grids and associated nested dyadic grids
Legendre-Gauss-Lobatto grids and associated nested dyadic grids
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Legendre-Gauss-Lobatto 网格和关联的嵌套二进网格
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
W. Dahmen
中科院分区：
文献类型：
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作者：
Kolja Brix;C. Canuto;W. Dahmen
Legendre-Gauss-Lobatto (LGL) 网格在偏微分方程数值求解的节点谱方法中发挥着关键作用。它们不仅提供有效的高阶求积规则，而且还产生范数等价，最终可能导致高阶方法中的有效预处理技术。不幸的是，不同程度的 LGL 网格不是嵌套的，这严重阻碍了充分利用这些概念的潜力。这一方面影响在应用辅助空间方法作为主要预处理范式时合适的辅助空间的选择和分析，另一方面影响辅助问题的有效解决。作为一种主要补救措施，我们考虑局部可比较网格大小的二元网格的某些嵌套层次结构，这些网格在某种意义上与 LGL 网格正确关联。它们的实际适用性需要对此类网格进行细致的分析，而这又依赖于 LGL 网格的许多细化属性。本文的中心目标就是推导出这些属性。这需要首先重新审视 LGL 网格的近亲属性，随后将其用于开发 LGL 网格的精细分析。这些结果使我们能够导出相关二元网格的相关属性。
Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive just these properties. This requires first revisiting properties of close relatives to LGL grids which are subsequently used to develop a refined analysis of LGL grids. These results allow us then to derive the relevant properties of the associated dyadic grids.
不连续伽辽金谱元方法的多级预处理第一部分：几何一致网格
影响因子：
2.1
作者：
Kolja Brix;Martin Campos Pinto;Claudio Canuto;Wolfgang Dahmen
通讯作者：
Wolfgang Dahmen