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：我们考虑将广义卷积求积法（gCQ）应用于求解一类重要的扇形问题。gCQ是对Lubich卷积求积法（CQ）的推广，允许使用可变步长。现有gCQ的稳定性与收敛性理论需要对数据提出非现实的规则性假设，而这些假设在诸多重要应用（如次扩散方程近似）中无法满足。众所周知，对于非光滑数据，采用均匀步长的原始CQ在奇点附近会出现阶数降低现象。我们将gCQ的分析推广至满足现实规则性假设的数据，并给出了任意时间点序列上稳定性与收敛性的充分条件。我们特别考虑了分级网格情形，并根据数据行为展示了如何最优选择网格参数。gCQ方法的一个重要优势在于其能实现快速且低内存消耗的计算实现。本文描述了快速无记忆化gCQ的实现方式，并通过多个数值实验验证了理论结果。