دانلود رایگان مقاله پیچیدگی اصلاح سلسله مراتبی برای طبقه از پیکربندی شبکه مجاز

Abstract

An adaptive isogeometric method based on d-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop ( Buffa and Giannelli, 2016). In this paper we provide a complexity estimate, i.e., an estimate on how the number of mesh elements grows with respect to the number of elements that are marked for refinement by the adaptive strategy. Our estimate is in the line of the similar ones proved in the context of adaptive finite element methods.

4. Conclusions

We developed a complexity estimate which states that the ratio between the refined elements and the marked elements along the refinement history stays bounded if refinement is performed as proposed in (Buffa and Giannelli, 2016). In particular, this estimate guarantees that if the refinement routine is applied very often in the same location (e.g., for resolving a singularity), then it will asymptotically remain local. Note that for a single refinement step, a uniform (with constants independent on the level) estimate bounding the number of refined elements in terms of the marked ones is not possible (Nochetto and Veeser, 2012). Our work paves the way to the analysis of optimal convergence of the adaptive strategy proposed in (Buffa and Giannelli, 2016) that will be addressed in further studies (Buffa and Giannelli, in preparation).