Isogeometric Analysis (IgA) is a spline based approach to the numerical solution of partial differential equations. There are two major issues that IgA was designed to address. The first issue is the exact representation of domains stemming 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 numbers 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. In the last decades, a zoo of local refinement strategies for splines has been developed. The authors think that many of these approaches are a burden to implement efficiently and impede the utilization of recent advances that rely on tensor-product splines, e.g., concerning matrix assembly and preconditioning. The implementation seems to be particularly cumbersome in the context of multi-patch IgA. Our approach is to moderately increase the number of patches and to utilize different grid sizes on different patches. 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.

翻译：等几何分析（Isogeometric Analysis, IgA）是一种基于样条求解偏微分方程数值解的途径。IgA旨在解决两大关键问题：其一，源于计算机辅助设计（CAD）软件生成的几何域需精确表示。在实践中，这只能通过多片IgA实现，且常需结合裁剪或类似技术。其二，实现高阶离散化（通过提升样条阶数）时，其自由度数量需与低阶方法相当。高阶方法若要充分发挥潜力，待求解问题需具备充分光滑性；否则需采用自适应方法。过去数十年间，学界已发展出多种样条局部加密策略。作者认为，多数方法在实现效率上存在负担，并阻碍了依赖张量积样条（如矩阵组装与预条件处理）等最新进展的推广应用。尤其在多片IgA框架下，其实现的复杂程度更为突出。本文提出一种适度增加片数并在不同片上采用不同网格尺寸的方法。该方法可复用现有代码库，达到与其他自适应方法相同的收敛速率，而自由度数量的增加幅度极小。