Unstructured spline spaces for isogeometric analysis based on spline manifolds

Based on Grimm and Hughes (1995) we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure which allows for the definition of function spaces such as, for instance, B-splines over multi-patch domains with extraordinary points or analysis-suitable unstructured T-splines. Within this framework, we generalize the concept of dual-compatible B-splines (developed for structured T-splines in Beirão da Veiga et al. (2013)). This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for h-refined meshes.