Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gau{\ss}-Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this paper we present a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods. Space decompositions are specified by collecting topological entities in a mesh (such as all vertices or facets) and applying a construction rule (such as taking all degrees of freedom in the cells around each entity). The software is implemented in PETSc and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. In turn, this allows for the very rapid development of fast solvers for difficult problems.

中文翻译：

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PCPATCH：用于多重网格松弛方法拓扑构建的软件

有效的松弛方法对于良好的多重网格收敛是必要的。对于很多方程，标准的 Jacobi 和 Gau{\ss}-Seidel 是不够的，需要更复杂的空间分解；例子包括半定项或鞍点结构的问题。在本文中，我们提出了一个统一的软件抽象，PCPATCH，用于多重网格松弛方法空间分解的拓扑构造。空间分解是通过收集网格中的拓扑实体（例如所有顶点或面）并应用构造规则（例如在每个实体周围的单元格中采用所有自由度）来指定的。该软件在 PETSc 中实现，仅通过在运行时改变求解器选项，就可以促进各种方案的优雅表达。反过来，