Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid.

中文翻译：

---

半结构化网格的非侵入式多网格

已经提出了用于分层混合网格（HHG）的多网格求解器，以促进高性能计算机体系结构的有效利用。这些HHG网格是通过均匀细化相对粗糙的完全非结构化网格而构造的。尽管HHG网格为非结构化应用程序提供了一定的灵活性，但是大多数多网格计算都可以使用高效的结构化网格思想和内核来完成。本文着重于推广HHG概念，以便将其应用于更广泛的计算科学家社区，并使现有应用程序更容易利用结构化多网格组件。具体来说，我们将结构化多重网格方法应用于明显更复杂的半结构化网格。进一步，我们说明了成熟的应用程序如何以相对非侵入性的方式采用半结构求解器。为此，我们提出了一个用于描述半结构化求解器的形式化数学框架。这种形式主义使我们能够精确定义关联的多重网格方法，并显示其与更传统的多重网格求解器的关系。此外，数学框架阐明了相关的软件设计和实现。数值实验突出了新求解器与经典多重网格的关系。我们还演示了与这种类型的半结构化多重网格相关的一般性和潜在的性能提升。这种形式主义使我们能够精确定义关联的多重网格方法，并显示其与更传统的多重网格求解器的关系。此外，数学框架阐明了相关的软件设计和实现。数值实验突出了新求解器与经典多重网格的关系。我们还演示了与这种类型的半结构化多重网格相关的一般性和潜在的性能提升。这种形式主义使我们能够精确定义关联的多重网格方法，并显示其与更传统的多重网格求解器的关系。此外，数学框架阐明了相关的软件设计和实现。数值实验突出了新求解器与经典多重网格的关系。我们还演示了与这种类型的半结构化多重网格相关的一般性和潜在的性能提升。