$C^s$-smooth isogeometric spline spaces over planar multi-patch parameterizations
(2008.06247)Abstract
The design of globally $Cs$-smooth ($s \geq 1$) isogeometric spline spaces over multi-patch geometries is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods [25,28] and [31-33] for the construction of $C1$-smooth and $C2$-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of $Cs$-smooth isogeometric multi-patch spline spaces of an arbitrary selected smoothness $s \geq 1$. More precisely, for any $s \geq 1$, we study the space of $Cs$-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular $Cs$-smooth subspace of the entire $Cs$-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this $Cs$-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the $Cs$-smooth spline functions to perform $L2$ approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed $Cs$-smooth subspace.
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