Isogeometric analysis with geometrically continuous functions on two-patch geometries

We study the linear space of $C^s$ ‐smooth isogeometric functions defined on a multi‐patch domain $\Omega \subset {‐{R}}^2$ . We show that the construction of these functions is closely related to the concept of geometric continuity of surfaces, which has originated in geometric design. More precisely, the $C^s$ ‐smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order ( $G^s$ ‐smoothness) of their graph surfaces. This motivates us to call them $C^s$ ‐smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of $C^1$ ‐smooth geometrically continuous isogeometric functions on bilinearly parameterized two‐patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing $L^2$ approximation and for solving Poisson’s equation and the biharmonic equation on two‐patch geometries are presented and indicate optimal rates of convergence.