An algebraic multigrid method based on an auxiliary topology with edge matrices

TL;DR

This paper presents a new algebraic multigrid method leveraging an auxiliary topology with edge matrices, improving convergence, parallelization, and communication efficiency for solving large linear systems from PDE discretizations.

Contribution

It introduces a novel energy-based coarsening criterion and prolongation construction that preserve kernel vectors and enhance parallel scalability.

Findings

Achieves guaranteed two-level convergence for certain elliptic PDE systems.

Demonstrates improved efficiency and robustness through numerical experiments.

Shows scalability and reduced communication in parallel implementations.

Abstract

This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two-level convergence even for systems of equations. This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.