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 faces) 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.
翻译:有效的放松方法对于良好的多格融合是必要的。 对于许多方程式来说,标准 Jacobi 和 Gaus 和 Gau ls}- Seidel 都不够充分,需要更复杂的空间分解;例子包括半无限期条件或马鞍点结构的问题。在本文中,我们提出了一个统一的软件抽象,即PCPATCH,用于多格放松方法的空间分解的地形构造。空间分解是通过在网格(如所有脊椎或面孔)中收集表层实体和适用建筑规则(如每个实体周围的细胞享有所有程度的自由)来指定的。软件在PETSC中应用,便于仅仅通过运行时的不同求解选项来优美地表达广泛的计划。这反过来又使得快速解决问题的快速解决方案能够快速发展。