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In this study we consider domains that are composed of an infinite sequenceof self-similar rings and corresponding finite element spaces over thosedomains. The rings are parameterized using piecewise polynomial ortensor-product B-spline mappings of degree $q$ over quadrilateral meshes. Wethen consider finite element discretizations which, over each ring, are mapped,piecewise polynomial functions of degree $p$. Such domains that are composed ofself-similar rings may be created through a subdivision scheme or from a scaledboundary parameterization.  We study approximation properties over such recursively parameterizeddomains. The main finding is that, for generic isoparametric discretizations(i.e., where $p=q$), the approximation properties always depend only on thedegree of polynomials that can be reproduced exactly in the physical domain andnot 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 meshsize, independent of the degree $p=q$. This has implications for subdivisionbased isogeometric analysis, which we will discuss in this paper.

Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysis