Analysis-suitable G^1 multi-patch parametrizations for C^1 isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to Cp-1 continuity within each patch. However, global continuity beyond C° on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C° at the patch interface. On such domains we study the h-refinement of C1-continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C1-continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C1 isogeometric spaces over analysis-suitable G1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C1 isogeometric spaces is prevented