In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a -smooth basis is easy to obtain, whereas -smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff–Love plate or shell formulation, using an isogeometric Galerkin method.
With the construction of so-called analysis-suitable (in short, AS-) parametrizations, as introduced in Collin et al. (2016), it is possible to construct isogeometric spaces which possess optimal approximation properties, cf. (Kapl et al., 2019). These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity and degree of the underlying spline space satisfy . The problem is that most complex geometries are not AS- geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate conditions following the basis construction from Kapl et al. (2017). For this reason, the defined function spaces are not exactly but only approximately.
We study the convergence behavior and define function spaces that converge optimally under -refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface.
