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We study the dimension and construct a basis for $C^1$-smooth isogeometricfunction spaces over two-patch domains. In this context, an isogeometricfunction is a function defined on a B-spline domain, whose graph surface alsohas a B-spline representation. We consider constructions along one interfacebetween two patches. We restrict ourselves to a special case of planar B-splinepatches of bidegree (p,p) with $p \geq 3$, so-called analysis-suitable $G^1$geometries, which are derived from a specific geometric continuity condition.This class of two-patch geometries is exactly the one which allows, undercertain additional assumptions, $C^1$ isogeometric spaces with optimalapproximation properties (Collin et al., 2016).  Such spaces are of interest when solving numerically fourth-order PDEproblems, such as the biharmonic equation, using the isogeometric method. Inparticular, we analyze the dimension of the $C^1$-smooth isogeometric space andpresent an explicit representation for a basis of this space. Both thedimension of the space and the basis functions along the common interfacedepend on the considered two-patch parameterization. Such an explicit, geometrydependent basis construction is important for an efficient implementation ofthe isogeometric method. The stability of the constructed basis is numericallyconfirmed for an example configuration.

Dimension and basis construction for analysis-suitable $G^1$ two-patch parameterizations