A meshless geometric multigrid (GMG) method based on a node-coarsening algorithm is proposed in the context of finite element method (FEM) with unstructured grids consisting of linear elements. Unlike the existing GMG methods, the present method does not require the generation of a sequence of coarse grids so that all the problems related to coarse-grid generation can be eliminated. Instead, only the sets of nodes in coarse levels are constructed for multigrid computation from the finest grid by using the node-coarsening algorithm that can be employed for any kind of a 2D/3D unstructured grid as well as a hybrid grid on the finest level. The implementation of the present coarsening algorithm is simple in the sense that the boundary information of the finest grid is not required. A searching algorithm to calculate the area/volume-shape function of the finite element method is also proposed to derive an operator for linear interpolation of multigrid computation. We have successfully validated the proposed method for various 2D/3D benchmark problems by showing that the elapsed time of the present GMG method is linearly proportional to the number of unknowns of a linear system of equations. We have also confirmed that the proposed method is nearly as efficient as the grid-based MG method based on a sequence of coarse grids in terms of CPU time. Lastly, we have successfully validated the meshless GMG method by solving an elliptic equation formulated by the finite volume method on a complicated 3D geometry filled with a hybrid unstructured mesh.

在有限元方法（FEM）的背景下，提出了一种基于节点粗化算法的无网格几何多网格（GMG）方法，该方法具有由线性元素组成的非结构化网格。与现有的GMG方法不同，本方法不需要生成一系列的粗网格，从而可以消除与生成粗网格有关的所有问题。取而代之的是，通过使用节点粗化，只能从最精细的网格中构建粗糙级别的节点集，以进行多网格计算该算法可用于任何种类的2D / 3D非结构化网格以及最高级的混合网格。在不需要最细网格的边界信息的意义上，本粗化算法的实现是简单的。还提出了一种计算有限元方法的面积/体积形状函数的搜索算法，以推导用于多网格计算的线性插值的算子。通过显示当前GMG方法的经过时间与线性方程组的未知数成线性比例，我们已经成功地验证了针对各种2D / 3D基准问题的建议方法。我们也已经证实，在CPU时间方面，所提出的方法与基于一系列粗网格的基于网格的MG方法几乎一样有效。最后，