A triangulation of a polytope into simplices is refined recursively. In every refinement round, some simplices which have been marked by an external algorithm are bisected and some others around also must be bisected to retain regularity of the triangulation. The ratio of the total number of marked simplices and the total number of bisected simplices is bounded from above. Binev, Dahmen and DeVore proved under a certain initial condition a bound that depends only on the initial triangulation. This thesis proposes a new way to obtain a better bound in any dimension. Furthermore, the result is proven for a weaker initial condition, invented by Alk\"amper, Gaspoz and Kl\"ofkorn, who also found an algorithm to realise this condition for any regular initial triangulation. Supposably, it is the first proof for a Binev-Dahmen-DeVore theorem in any dimension with always practically realiseable initial conditions without an initial refinement. Additionally, the initialisation refinement proposed by Kossaczk\'y and Stevenson is generalised, and the number of recursive bisections of one single simplex in one refinement round is bounded from above by twice the dimension, sharpening a result of Gallistl, Schedensack and Stevenson.

翻译：对多面体的单纯形三角剖分进行递归加密。在每一轮加密中，由外部算法标记的部分单纯形被对半剖分，而周围的其他单纯形也必须被剖分以保持三角剖分的正则性。被标记的单纯形总数与被剖分的单纯形总数之比存在上界。Binev、Dahmen和DeVore在特定初始条件下证明了一个仅依赖于初始三角剖分的上界。本文提出了一种在任意维度上获取更优上界的新方法。此外，该结果在由Alkämper、Gaspoz和Klöfkorn提出的更弱初始条件下得到证明，他们同时发现了一种算法可在任意正则初始三角剖分上实现该条件。这可能是首个在任意维度且无需初始加密即可实现实际可行初始条件的Binev-Dahmen-DeVore定理证明。此外，本文推广了Kossaczký和Stevenson提出的初始化加密方法，并将单轮加密中单个单纯形的递归剖分次数上界改进为维数的两倍，强化了Gallistl、Schedensack和Stevenson的结论。