Motivated by the promising recent results concerning the construction of smooth isogeometric functions on multi-patch domains on bilinearly parameterized domains [14] or reparameterizations of more general domains [5], which, however, impose quite restrictive assumptions on the underlying domain, we propose two approaches to construct spaces G~(1,ε)_h of approximately C~1-smooth isogeometric functions on general two-patch domains. The main idea is to work with C~0-continuous functions and to bound the jump of their gradients across the interface between neighboring patches. The constructions are based on two suitably chosen bilinear forms B_1 and B_2 and their eigenstructures, which lead to different bounds on the gradient jumps, respectively. We show that while the gradient jumps of the functions based on B_1 fulfill a stricter bound, the functions themselves do not realize optimal convergence rates. Numerical experiments suggest that the functions based on B_2 reach the optimal approximation order for solving second order problems. Furthermore, they are smooth enough to solve higher order problems such as the biharmonic equation. However, the bound on their gradient jump is mesh-size dependent.
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机译：有关在双线性参数化结构域的多贴片域上的平滑异步函数的近期建设的有希望的最新结果的推动是更多的，或者更普通域的Reparameterations [5]，但是，我们提出了对底层领域的限制性假设。在一般的双贴片域上构造空间G〜（1，ε）_h的两个方法，大约是C〜1平滑的异构函数。主要思想是使用C〜0连续功能，并在相邻修补程序之间的接口中绑定其渐变跳转。 The constructions are based on two suitably chosen bilinear forms B_1 and B_2 and their eigenstructures, which lead to different bounds on the gradient jumps, respectively.我们表明，虽然基于B_1的功能的渐变跳跃实现更长的绑定，但功能本身本身不会实现最佳收敛速率。数值实验表明，基于B_2的功能达到了解决二阶问题的最佳逼近顺序。此外，它们足够平滑以解决比乐队方程等高阶问题。但是，梯度跳跃的界限是依赖的网格尺寸。
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