We consider the application of the generalized Convolution Quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments.

翻译：本文探讨了广义卷积求积法在近似求解一类重要扇形问题中的应用。广义卷积求积法是对Lubich卷积求积法的推广，允许采用变步长。现有关于广义卷积求积法的稳定性和收敛性理论要求数据满足不切实际的正则性假设，这在许多重要应用场景（如亚扩散方程的近似求解）中并不成立。众所周知，对于光滑性不足的数据，采用均匀步长的原始卷积求积法会在奇点附近出现阶数退化现象。本文将广义卷积求积法的分析推广至满足实际正则性假设的数据，并为任意时间点序列提供了稳定性和收敛性的充分条件。我们特别研究了分级网格的情形，并展示了如何根据数据特性进行最优网格选择。广义卷积求积法的一个重要优势在于其可实现快速且内存优化的计算。本文阐述了快速无记忆广义卷积求积法的实现方式，并通过多组数值实验验证了理论结果。