Approximation properties of multi-patch $C^1$ isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. This is achieved by $p$-degree splines (and extensions, such as NURBS) that are globally up to $C^{p-1}$-continuous in each patch. However, global continuity beyond $C^0$ on so-called multi-patch geometries poses some significant difficulties. In this work, we consider multi-patch domains that have a parametrization which is only $C^0$ at the patch interface. On such domains we study the $h$-refinement of $C^1$-continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the $C^1$-continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recent studies by Kapl et al. have given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and $C^1$ splines of polynomial degree at least 3. This is a key result and the starting point of our study. We introduce analysis-suitable $G^1$-continuous geometry parametrizations, a class of parametrizations that includes bilinears. We analyze the structure of $C^1$ isogeometric spaces and, by theoretical results and numerical testing, infer that analysis-suitable $G^1$ geometry parametrizations are the ones that allow optimal convergence of $C^1$ isogeometric spaces (under conditions on the continuity along the interface). Beyond analysis-suitable $G^1$ parametrizations optimal convergence is prevented.