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

Thomas Takacs (JKU Linz)

Abstract.

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, as will we point out in this work.