A meshless geometric multigrid method based on a node-coarsening algorithm for the linear finite element discretization

Abstract

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.