负序索伯列夫空间的多层分解与范数
Multilevel decompositions and norms for negative order Sobolev spaces
摘要：
我们认为多级分解的分段常数的单纯网格是稳定的$H^{-s}$为$s\in (0,1)$。在均匀和局部加细网格的情况下给出了证明。我们的研究结果可以应用于定义本地多级对角预条件，导致有界的条件数（独立的网格大小和水平），并具有最佳的计算复杂性。此外，我们还讨论了基于局部（准）投影算子的多级范数，该算子可以有效地评价负序Sobolev范数。数值例子和讨论的几个扩展和应用结束这篇文章。
英文摘要：
We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in $H^{-s}$ for $s\in (0,1)$. Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.
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