A Nearly Optimal Multigrid Method for General Unstructured Grids

TL;DR

This paper introduces a nearly optimal multigrid method for solving elliptic PDEs on general unstructured shape-regular grids, achieving near-linear complexity and proven convergence rates.

Contribution

It presents a novel auxiliary space multigrid method using a cluster tree for unstructured grids, with proven convergence and near-linear complexity.

Findings

Convergence rate of 1 - O(1/ log N) for the multigrid preconditioned CG method.

Total computational complexity of O(N log N).

Numerical experiments confirm theoretical bounds.

Abstract

In this paper, we develop a multigrid method on unstructured shape-regular grids. For a general shape-regular unstructured grid of ${\cal O}(N)$ elements, we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique. Such a construction is realized by a cluster tree which can be obtained in ${\cal O}(N\log N)$ operations for a grid of $N$ elements. This tree structure in turn is used for the definition of the grid hierarchy from coarse to fine. For the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned CG for an elliptic PDE is $1 - {\cal O}({1}/{\log N})$. Numerical experiments confirm the theoretical bounds and show that the total complexity is in ${\cal O}(N\log N)$.