Isogeometric Analysis (IgA) is a spline-based approach to the numerical solution of partial differential equations. The concept of IgA was designed to address two major issues. The first issue is the exact representation of domains generated from Computer-Aided Design (CAD) software. In practice, this can be realized only with multi-patch IgA, often in combination with trimming or similar techniques. The second issue is the realization of high-order discretizations (by increasing the spline degree) with a number of degrees of freedom comparable to low-order methods. High-order methods can deliver their full potential only if the solution to be approximated is sufficiently smooth; otherwise, adaptive methods are required. A zoo of local refinement strategies for splines has been developed in the last decades. Such approaches impede the utilization of recent advances that rely on tensor-product splines, e.g., matrix assembly and preconditioning. We propose a strategy for adaptive IgA that utilizes well-known approaches from the multi-patch IgA toolbox: using tensor-product splines locally, but allow for unstructured patch configurations globally. Our approach moderately increases the number of patches and utilizes different grid sizes for each patch. This allows reusing the existing code bases, recovers the convergence rates of other adaptive approaches, and increases the number of degrees of freedom only marginally. We provide an algorithm for the computation of a global basis and show that it works in any case. Additionally, we give approximation error estimates. Numerical experiments illustrate our results.

中文翻译：

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多分辨率等几何分析 – 利用多面片结构实现高效自适应


等几何分析 （IgA） 是一种基于样条的方法，用于对偏微分方程进行数值求解。IgA 的概念旨在解决两个主要问题。第一个问题是计算机辅助设计 （CAD） 软件生成的域的准确表示。在实践中，这只能通过多斑块 IgA 来实现，通常与修剪或类似技术相结合。第二个问题是实现高阶离散化（通过增加样条度），其自由度数可与低阶方法相媲美。只有当要近似的解足够平滑时，高阶方法才能发挥其全部潜力;否则，需要 adaptive 方法。在过去的几十年中，已经开发了一个样条的局部细化策略动物园。这种方法阻碍了依赖于张量积样条的最新进展的利用，例如矩阵组装和预处理。我们提出了一种自适应 IgA 策略，该策略利用了多补丁 IgA 工具箱中众所周知的方法：在本地使用张量积样条，但允许全局使用非结构化补丁配置。我们的方法适度增加补丁的数量，并为每个补丁使用不同的网格大小。这允许重用现有的代码库，恢复其他自适应方法的收敛率，并且仅略微增加自由度的数量。我们提供了一种计算全局基的算法，并证明它在任何情况下都有效。此外，我们还给出了近似误差估计值。数值实验说明了我们的结果。