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：本文研究广义卷积求积法在近似求解一类重要扇形问题解中的应用。广义卷积求积法是对Lubich卷积求积法的推广，允许采用变步长计算。现有关于广义卷积求积法的稳定性与收敛性理论要求数据满足非现实的正则性假设，这类假设在许多重要应用场景（如亚扩散方程的数值逼近）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情形，并给出了任意时间点序列下稳定性与收敛性的充分条件。针对分级网格这一特殊情形，我们阐明了如何根据数据特性进行最优网格选取。广义卷积求积法的重要优势在于可实现快速且内存优化的算法实施。本文阐述了快速无记忆广义卷积求积法的实现方案，并通过系列数值实验验证了理论结果。