标题： 一种用于DPG系统矩阵的几何多重网格预条件策略 显示英文标题

标题： A Geometric Multigrid Preconditioning Strategy for DPG System Matrices

摘要： 不连续Petrov-Galerkin (DPG)方法由Demkowicz和Gopalakrishnan [15,17]提出，保证了解在能量范数下的最优性，并提供了便于自适应方案的多个特性。一个尚未普遍回答的关键问题是——尽管对于泊松方程有一些结果——如何最好地对DPG系统矩阵进行预处理，以便使用迭代求解器来解决大规模问题。在本文中，我们详细描述了一种使用几何多重网格对DPG系统矩阵进行预处理的策略，我们已将其作为Camellia [26]的一部分实现，并通过数值实验展示了其在几种变分公式中的有效性。我们观察到，在一些实验中，预处理器的行为与离散测试空间的丰富密切相关。我们包括了涉及带有悬挂节点的自适应网格的 lid-driven cavity flow 实验，证明了预处理器可以在具有挑战性的问题背景下应用。我们还包括了一个可扩展性研究，证明该方法及我们的实现能够很好地扩展到许多MPI进程。 显示英文摘要

摘要： The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [15,17] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. A key question that has not yet been answered in general - though there are some results for Poisson, e.g. - is how best to precondition the DPG system matrix, so that iterative solvers may be used to allow solution of large-scale problems. In this paper, we detail a strategy for preconditioning the DPG system matrix using geometric multigrid which we have implemented as part of Camellia [26], and demonstrate through numerical experiments its effectiveness in the context of several variational formulations. We observe that in some of our experiments, the behavior of the preconditioner is closely tied to the discrete test space enrichment. We include experiments involving adaptive meshes with hanging nodes for lid-driven cavity flow, demonstrating that the preconditioners can be applied in the context of challenging problems. We also include a scalability study demonstrating that the approach - and our implementation - scales well to many MPI ranks.