We consider the application of the generalized Convolution Quadrature (gCQ) of the first order to approximate fractional integrals and associated fractional diffusion equations. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) which allows for variable steps. In this paper we analyze the application of the gCQ to fractional integrals, with a focus on the low regularity case. It is well known that in this situation the original CQ presents an order reduction close to the singularity. Moreover, the available theory for the gCQ does not cover this situation. Here we deduce error bounds for a general time mesh. We show first order of convergence under much weaker regularity requirements than previous results in the literature. We also prove that uniform first order convergence is achievable for a graded time mesh, which is appropriately refined close to the singularity, according to the order of the fractional integral and the regularity of the data. Then we study how to obtain full order of convergence for the application to fractional diffusion equations. For the implementation of this method, we use fast and oblivious quadrature and present several numerical experiments to illustrate our theoretical results.
翻译:我们考虑对近似分构件和相关的分片扩散方程式应用第一种统称的全变夸度(gCQ) 。 gCQ 是允许不同步骤的Lubich 常规度(CQ) 的统称。 在本文中,我们分析了对分构件的应用 gCQ 的统称, 重点是低规律性案例。 众所周知, 在这种情形下, 原 CQ 的原CQ 显示的顺序减少接近于单一性。 此外, GCQ 的现有理论并不涵盖这种情况。 我们在此推导出一个一般时间网格的错误界限。 我们显示了在比文献中以往结果差得多的常规性要求下的第一个趋同顺序。 我们还证明, 在一个分级的网格中, 统一的第一顺序的趋同性是可以实现的, 并且按照分集的顺序和数据规律性来适当改进的。 然后我们研究如何在应用分构件式方程式时获得完全的趋同性。 为了实施这一方法, 我们使用快速和模糊的理论性实验来说明我们目前的几度。