  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 $\widetilde{p} \leq p$ with regularity
$\widetilde{r} = \widetilde{p}-1 \geq r$ in all other parts of the domain. Our
proposed approach relies on the technique [35], 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 [35], 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 [33,35], 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.
