A Cs-smooth mixed degree and regularity isogeometric spline space over planar multi-patch domains

Abstract

We construct over a given bilinear multi-patch domain a novel $C^s$-smooth mixed degree and regularity isogeometric spline space, which possesses the degree $p=2s+1$ and regularity $r=s$ in a small neighborhood around the edges and vertices, and the degree $\tilde{p}\le p$ with regularity $\tilde{r}=\tilde{p}-1\ge r$ in all other parts of the domain. Our proposed approach relies on the technique Kapl and Vitrih (2021), which requires for the $C^s$-smooth isogeometric spline space a degree at least $p=2s+1$ on the entire multi-patch domain. Similar to Kapl and Vitrih (2021), the $C^s$-smooth mixed degree and regularity spline space is generated as the span of basis functions that correspond to the individual patches, edges and vertices of the domain. The reduction of degrees of freedom for the functions in the interior of the patches is achieved by introducing an appropriate mixed degree and regularity underlying spline space over ${[0,1]}^2$ to define the functions on the single patches. We further extend our construction with a few examples to the class of bilinear-like $G^s$ multi-patch parameterizations (Kapl and Vitrih (2018); Kapl and Vitrih (2021)), which enables the design of multi-patch domains having curved boundaries and interfaces. Finally, the great potential of the $C^s$-smooth mixed degree and regularity isogeometric spline space for performing isogeometric analysis is demonstrated by several numerical examples of solving two particular high order partial differential equations, namely the biharmonic and triharmonic equation, via the isogeometric Galerkin method.