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

Jiayin Li, Jinbiao Wu, Wenqian Zhang, Jiawen Liu

TL;DR

The paper addresses solving large sparse linear systems from linear elasticity by introducing a vertex-based auxiliary space multigrid (V-ASMG) preconditioner for PCG. The core idea is to connect an unstructured fine grid to structured coarse grids via an auxiliary region-tree and vertex-based representatives, enabling simple bilinear interpolation for prolongation/restriction and a two-level MG flow that yields a uniformly bounded condition number. Theoretical results establish κ(B_h A_h) = O(1) under standard assumptions, and extensive 2D and 3D numerical experiments demonstrate that V-ASMG significantly reduces iteration counts and total solve times compared with classical AMG and B-ASMG, albeit with higher setup costs due to region-tree construction. The method’s simplicity, scalability, and strong convergence suggest broad applicability to linear elasticity and potentially other PDEs, offering a practical acceleration for large-scale finite element systems.

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.

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

TL;DR

The paper addresses solving large sparse linear systems from linear elasticity by introducing a vertex-based auxiliary space multigrid (V-ASMG) preconditioner for PCG. The core idea is to connect an unstructured fine grid to structured coarse grids via an auxiliary region-tree and vertex-based representatives, enabling simple bilinear interpolation for prolongation/restriction and a two-level MG flow that yields a uniformly bounded condition number. Theoretical results establish κ(B_h A_h) = O(1) under standard assumptions, and extensive 2D and 3D numerical experiments demonstrate that V-ASMG significantly reduces iteration counts and total solve times compared with classical AMG and B-ASMG, albeit with higher setup costs due to region-tree construction. The method’s simplicity, scalability, and strong convergence suggest broad applicability to linear elasticity and potentially other PDEs, offering a practical acceleration for large-scale finite element systems.

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 , where is the number of the given original grid vertices and is the power of the ratio of the maximum distance to minimum distance 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 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.
Paper Structure (14 sections, 6 theorems, 50 equations, 22 figures, 16 tables)

This paper contains 14 sections, 6 theorems, 50 equations, 22 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GS method
  4. PCG method
  5. V-ASMG method
  6. Grid hierarchy relationship
  7. The relationship between the fine grid and the coarse grid of the first layer
  8. The relationship between the coarse grids
  9. Prolongation/restriction operator
  10. V-ASMG preconditioner
  11. Numerical examples
  12. The two-dimensional problems
  13. The three-dimensional problems
  14. Concluding remarks

Key Result

Theorem 1

Assume that the maximum distance $d_{max}$ and minimum distance $d_{min}$ between the vertexes of the given initial fine grid are satisfied where $N$ represents the number of fine grid vertexes, q is a small number, such as q=2, and $d_{max}$ is also called the diameter of the region $\Omega$, denoted by $\mathrm{diam}(\Omega)=d_{max}$. Based on the above assumption, the computational complexity

Figures (22)

  • Figure 1: The cycle modes of different grid layers of MG method.
  • Figure 2: The profile of unstructured triangulation partition $\mathcal{T}_h$.
  • Figure 15: The profile of structured coarse grids of each layer.
  • Figure 16: The profile of structured fine and coarse grids and their weights.
  • Figure 17: Round-hole plate tensile problem.
  • ...and 17 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 3 more