In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline mappings of degree $q$ over quadrilateral meshes. We then consider finite element discretizations which, over each ring, are mapped, piecewise polynomial functions of degree $p$. Such domains that are composed of self-similar rings may be created through a subdivision scheme or from a scaled boundary parameterization. We study approximation properties over such recursively parameterized domains. The main finding is that, for generic isoparametric discretizations (i.e., where $p=q$), the approximation properties always depend only on the degree of polynomials that can be reproduced exactly in the physical domain and not on the degree $p$ of the mapped elements. Especially, in general, $L^\infty$-errors converge at most with the rate $h^2$, where $h$ is the mesh size, independent of the degree $p=q$. This has implications for subdivision based isogeometric analysis, which we will discuss in this paper.

翻译：本研究考虑由无限序列的自相似环组成的域以及这些域上的相应有限元空间。这些环通过四边形网格上的分片多项式或次数为$q$的张量积B样条映射进行参数化。随后，我们考虑在每个环上映射的、次数为$p$的分片多项式函数的有限元离散化。此类由自相似环组成的域可通过细分方案或缩放边界参数化生成。我们研究这类递归参数化域上的逼近性质。主要发现是，对于一般的等参离散化（即$p=q$的情况），逼近性质始终仅取决于物理域中能够精确再现的多项式次数，而非映射单元的次数$p$。特别地，一般情况下，$L^\infty$误差的收敛速率最多为$h^2$，其中$h$为网格尺寸，且该速率与次数$p=q$无关。这一结论对基于细分的等几何分析具有重要意义，本文也将对此进行讨论。