Approximation properties of multi-patch C1 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 Cp?1-continuous in each patch. However, global continuity beyond C0 on so-called multi-patch geometries poses some significant diculties. In this work, we consider multi-patch domains that have a parametrization which is only C0 at the patch interface. On such domains we study the h-renement 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 [20] has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and C1 splines of polynomial degree at least 3. This is a key result and the starting point of our study. We introduce analysis-suitable G1-continuous geometry parametrizations, a class of parametrizations that includes bilinears. We analyze the structure of C1 isogeometric spaces and, by theoretical results and numerical testing, infer that analysis-suitable G1 geometry parametrizations are the ones that allow optimal convergence of C1 isogeometric spaces (under conditions on the continuity along the interface). Beyond analysis-suitable G1 parametrizations optimal convergence is prevented.