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

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 ∞ -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. • We study approximation with piecewise polynomials over domains composed of self-similar rings. • We prove that convergence rates are determined by polynomial reproduction in physical coordinates. • The results cover polar and scaled boundary parameterization and subdivision surfaces. • We provide an explanation why convergence rates for subdivision based discretizations are often suboptimal.