In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a $C^0$-smooth basis is easy to obtain, whereas $C^1$-smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff–Love plate or shell formulation, using an isogeometric Galerkin method.

With the construction of so-called analysis-suitable $G^1$ (in short, AS-$G^1$) parametrizations, as introduced in Collin et al. (2016), it is possible to construct $C^1$ isogeometric spaces which possess optimal approximation properties, cf. (Kapl et al., 2019). These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity $r$ and degree $p$ of the underlying spline space satisfy $1\le r\le p-2$. The problem is that most complex geometries are not AS-$G^1$ geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate $C^1$ conditions following the basis construction from Kapl et al. (2017). For this reason, the defined function spaces are not exactly $C^1$ but only approximately.

We study the convergence behavior and define function spaces that converge optimally under $h$-refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface.

在本文中，我们开发和研究了用于两面片域上的等几何分析的近似平滑基构造。等几何分析的一个关键要素是它允许一个补丁内的高阶平滑度。但是，为了表示复杂的几何图形，需要多面片构造。在这种情况下，一个$C^0$- 光滑基很容易获得，而 $C^1$-平滑等几何函数需要特殊的构造。在使用等几何 Galerkin 方法求解数值四阶 PDE 问题（例如双调和方程和 Kirchhoff-Love 板或壳公式）时，此类空间很重要。

随着所谓的分析适合 $G^1$ （简而言之，AS-$G^1$) 参数化，如 Collin 等人介绍的那样。(2016), 可以构建$C^1$具有最佳逼近性质的等几何空间，参见。（Kapl 等人，2019 年）。这些几何形状需要满足沿界面的某些约束，另外还需要规则性 $r$ 和学位 $磷$ 基础样条空间的满足 $1\le r\le 磷-2$. 问题是大多数复杂的几何图形都不是 AS-$G^1$几何形状。因此，我们通过强制近似来定义等几何空间的基函数$C^1$遵循 Kapl 等人的基础构建的条件。(2017)。因此，定义的函数空间并不完全是$C^1$ 但只是大约。

我们研究收敛行为并定义在以下条件下最佳收敛的函数空间 $H$-refinement，通过局部引入更高多项式次数和更低正则性的函数。在具有非平凡界面的域上执行的几个数值测试中，收敛速度是最佳的。虽然扩展到更一般的多补丁域是可能的，但我们将自己限制在两个补丁的情况下，并专注于单个接口上的构建。