Two new relaxation schemes are proposed for the smoothing step in the geometric multigrid solution of PDEs on 2D and 3D stretched structured grids. The new schemes are characterized by efficient line relaxation on branched sets of lines of alternating colour, where the lines are constructed to be everywhere orthogonal to the local direction of maximum grid clustering. Tweed relaxation is best suited for grid clustering near the boundaries of the computational domain, whereas wireframe relaxation is best suited for grid clustering near the centre of the computational domain. On strongly stretched grids of these types, multigrid leveraging these new smoothing schemes significantly outperforms multigrid based on other leading relaxation schemes, such as checkerboard and alternating-direction zebra relaxation, for the numerical solution of large linear systems arising from the discretization of elliptic PDEs.

翻译：在2D和3D拉伸结构网格的PDE的几何多格解决方案中,提出了两个新的松动计划,以平滑步骤解决PDE的2D和3D拉伸结构网格。新的计划的特点是在交替颜色的分支线条上有效松动线条,这些线条的构造无处不在,与最大网格集群的本地方向一致。Tweed 松动最适合在计算域的边界附近进行网格集群,而线框松动最适合在计算域中心附近进行网格集群。在这种高度拉长的网格上,多格利用这些新的光滑动计划大大超过基于其他主要放松计划(例如支票板和交替方向斑马松动)的多格,以便从数字上解决因椭圆形PDE离散而产生的大型线性系统。