We deal with accelerating the solution of a sequence of large linear systems solved by an iterative Krylov subspace method. The sequence originates from time-stepping within a simulation of an unsteady incompressible flow. We apply a pressure correction scheme, and we focus on the solution of the Poisson problem for the pressure corrector. Its scalable solution presents the main computational challenge in many applications. The right-hand side of the problem changes in each time step, while the system matrix is constant and symmetric positive definite. The acceleration techniques are studied on a particular problem of flow around a unit sphere. Our baseline approach is based on a parallel solution of each problem in the sequence by nonoverlapping domain decomposition method. The interface problem is solved by the preconditioned conjugate gradient (PCG) method with the three-level BDDC preconditioner. Three techniques for accelerating the solution are gradually added to the baseline approach. First, the stopping criterion for the PCG iterations is studied. Next, deflation is used within the conjugate gradient method with several approaches to Krylov subspace recycling. Finally, we add the adaptive selection of the coarse space within the three-level BDDC method. The paper is rich in experiments with careful measurements of computational times on a parallel supercomputer. The combination of the acceleration techniques eventually leads to saving about one half of the computational time.

翻译：本文研究如何加速由迭代Krylov子空间方法求解的一系列大型线性系统的计算过程。该序列源于非定常不可压缩流动模拟中的时间步进计算。我们采用压力修正格式，并重点研究压力修正项泊松问题的求解。该问题的可扩展求解是许多应用中的核心计算挑战。该问题的右端项在每一时间步发生变化，而系统矩阵保持恒定且对称正定。加速技术的研究以单位球绕流这一具体问题为对象。我们的基准方法基于非重叠区域分解法对序列中各问题进行并行求解，其中界面问题采用带三级BDDC预条件子的预条件共轭梯度（PCG）方法求解。我们在基准方法中逐步引入三种加速技术：首先研究PCG迭代的收敛准则；其次在共轭梯度法中结合多种Krylov子空间循环策略进行收缩处理；最后在三级BDDC方法中引入粗空间的自适应选择机制。本文通过大量实验，在并行超级计算机上精确测量计算时间。综合运用这些加速技术最终可节省约一半的计算时间。