论文标题
非平滑部门问题的广义卷积正交
Generalized Convolution Quadrature for non smooth sectorial problems
论文作者
论文摘要
我们考虑使用广义卷积正交（GCQ）的应用来近似重要的部门问题的解决方案。 GCQ是Lubich卷积正交（CQ）的概括，该卷积正交（CQ）允许进行可变步骤。 GCQ的可用稳定性和收敛理论需要对数据进行非现实的规律性假设，而这些假设在许多感兴趣的应用程序中不存在，例如次扩散方程的近似。众所周知，对于非平滑数据，原始CQ（均匀步骤）呈现出接近奇异性的订单降低。我们将GCQ的分析推广到满足现实规律性假设的数据，并为任意时间点的稳定性和收敛提供了足够的条件。我们考虑了分级网格的特定情况，并根据数据的行为来显示如何最佳选择它们。 GCQ方法的一个重要优点是它允许快速和内存减少实现。我们描述了如何通过几个数值实验来实现快速和遗忘的GCQ并说明我们的理论结果。
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.
