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：我们考虑应用广义卷积求积法（gCQ）逼近一类重要扇形问题的解。gCQ是Lubich卷积求积法（CQ）的推广，允许变步长。现有gCQ的稳定性和收敛性理论需要对数据做出不现实的规则性假设，这在许多实际应用（如次扩散方程逼近）中无法满足。众所周知，对于非光滑性充分的数据，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的分析推广到满足实际规则性假设的数据，并给出在任意时间点序列上实现稳定性和收敛性的充分条件。我们特别考虑分级网格情形，展示如何根据数据行为优化选择网格。gCQ方法的一个重要优势在于可实现快速且低内存消耗的数值实现。我们描述了快速无记忆gCQ的实现方法，并通过多项数值实验验证理论结果。