Dimension and basis construction for analysis-suitable $G^1$ two-patch parameterizations
Provides a geometry-dependent basis construction for $C^1$ isogeometric spaces on two-patch domains, which is important for efficient implementation of isogeometric methods for fourth-order PDEs.
The paper analyzes the dimension and constructs an explicit basis for $C^1$-smooth isogeometric function spaces over analysis-suitable $G^1$ two-patch domains of bidegree (p,p) with p≥3. The basis depends on the parameterization and is numerically stable for an example configuration.
We study the dimension and construct a basis for $C^1$-smooth isogeometric function spaces over two-patch domains. In this context, an isogeometric function is a function defined on a B-spline domain, whose graph surface also has a B-spline representation. We consider constructions along one interface between two patches. We restrict ourselves to a special case of planar B-spline patches 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, under certain additional assumptions, $C^1$ isogeometric spaces with optimal approximation properties (Collin et al., 2016). Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation, using the isogeometric method. In particular, we analyze the dimension of the $C^1$-smooth isogeometric space and present an explicit representation for a basis of this space. Both the dimension of the space and the basis functions along the common interface depend on the considered two-patch parameterization. Such an explicit, geometry dependent basis construction is important for an efficient implementation of the isogeometric method. The stability of the constructed basis is numerically confirmed for an example configuration.