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

## TL;DR

This paper investigates the dimension and basis construction for $C^1$-smooth isogeometric function spaces over two-patch geometries, specifically analysis-suitable $G^1$ planar B-spline patches, to facilitate efficient numerical solutions of fourth-order PDEs.

## Contribution

It provides an explicit, geometry-dependent basis construction and analyzes the dimension of $C^1$ isogeometric spaces on analysis-suitable $G^1$ two-patch geometries.

## Key findings

- Explicit basis functions depend on the geometry.
- Numerical confirmation of basis stability.
- Dimension analysis aligns with theoretical expectations.

## Abstract

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.