An isogeometric C 1 subspace on unstructured multi-patch planar domains

Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. Typically, quadrilateral patches are adopted in both frameworks. We consider the particular class of multi-patch parametrizations that are analysis-suitable G(1) (AS-G(1)), which is a specific geometric continuity definition which allows to construct, on the multi-patch domain, C-1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016). It was demonstrated in Kapl et al. (2018) that AS-G(1) multi-patch parametrizations are suitable for modeling complex planar multi-patch domains. We construct a local basis, and an associated dual basis, for a specific C-1 isogeometric spline space A over a given AS-G(1) multi-patch parametrization. The space A is C-1 across interfaces and C-2 at all vertices, and is therefore a subspace of the entire C-1 isogeometric space V-1. At the same time, A allows optimal approximation of traces and normal derivatives along the interfaces and reproduces all derivatives up to second order at the vertices. In contrast to V-1, the dimension of A does not depend on the domain parametrization. This paper also contains numerical experiments which exhibit the optimal approximation order in L-2 and L-infinity of the isogeometric space A and demonstrate the applicability of our approach for isogeometric analysis.