Construction of approximate $C^1$ bases for isogeometric analysis on two-patch domains

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 $C^0$-smooth basis is easy to obtain, whereas $C^1$-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 $G^1$ (in short, AS-$G^1$) parametrizations, as introduced in (Collin, Sangalli, Takacs; CAGD, 2016), it is possible to construct $C^1$ isogeometric spaces which possess optimal approximation properties. These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity $r$ and degree $p$ of the underlying spline space satisfy $1 \leq r \leq p-2$. The problem is that most complex geometries are not AS-$G^1$ geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate $C^1$ conditions following the basis construction from (Kapl, Sangalli, Takacs; CAGD, 2017). For this reason, the defined function spaces are not exactly $C^1$ but only approximately. We study the convergence behavior and define function spaces that converge optimally under $h$-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.