Isogeometric analysis with geometrically continuous functions on two-patch geometries

We study the linear space of CsCs-smooth isogeometric functions defined on a multi-patch domain  Ω⊂R2Ω⊂R2. 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 CsCs-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (GsGs-smoothness) of their graph surfaces. This motivates us to call them CsCs-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C1C1-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 L2L2 approximation and for solving Poisson’s equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.