Vertex-based auxiliary space multigrid method and its application to linear elasticity equations

TL;DR

This paper introduces a vertex-based auxiliary space multigrid method as an efficient preconditioner for solving large sparse linear systems from linear elasticity equations, emphasizing simplicity and strong convergence.

Contribution

The paper proposes a novel V-ASMG method utilizing an auxiliary region-tree based on grid vertices, improving simplicity and convergence over previous approaches.

Findings

The V-ASMG method converges quickly to a residual of 10^{-6}.

Numerical experiments confirm the method's effectiveness in 2D and 3D.

The approach simplifies construction and generalization to other problems.

Abstract

In this paper, a vertex-based auxiliary space multigrid(V-ASMG) method as a preconditioner of the PCG method is proposed for solving the large sparse linear equations derived from the linear elasticity equations. The main key of such V-ASMG method lies in an auxiliary region-tree structure based on the geometrically regular subdivision. The computational complexity of building such a region-tree is $\mathcal{O}\left(q N\log_2 N\right)$, where $N$ is the number of the given original grid vertices and $q$ is the power of the ratio of the maximum distance $d_{max}$ to minimum distance $d_{min}$ between the given original grid vertices. The process of constructing the auxiliary region-tree is similar to the method in [17], but the selection of the representative points is changed. To be more specific, instead of choosing the barycenters, the correspondence between each grid layer is constructed based on the position relationship of the grid vertices. There are two advantages for this approach: the first is its simplicity, there is no need to deal with hanging points when building the auxiliary region-tree, and it is possible to construct the restriction/prolongation operator directly by using the bilinear interpolation function, and it is easy to be generalized to other problems as well, due to all the information we need is only the grid vertices; the second is its strong convergence, the corresponding relative residual can quickly converge to the given tolerance(It is taken to be $10^{-6}$ in this paper), thus obtaining the desired numerical solution. Two- and three-dimensional numerical experiments are given to verify the strong convergence of the proposed V-ASMG method as a preconditioner of the PCG method.