A geometrically informed algebraic multigrid preconditioned iterative approach for solving high-order finite element systems

TL;DR

This paper introduces a geometrically informed algebraic multigrid (GIAMG) method that leverages geometric information for efficient high-order finite element system solutions, showing improved scalability and performance over existing AMG methods.

Contribution

The paper develops a novel GIAMG approach that incorporates geometric p-coarsening into AMG hierarchies, enhancing solver efficiency for high-order finite element problems.

Findings

Achieves mesh-independent convergence in 3D Helmholtz and flow problems.

Demonstrates excellent parallel scalability of GIAMG.

Outperforms existing AMG packages like Hypre and ML.

Abstract

Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is beneficial, especially for solving linear systems resulting from high-order finite element discretizations. High-order problems draw considerable interest to both the scientific and engineering communities, but lack efficient solvers, at least open-source codes, tailored for unstructured high-order discretizations targeting large-scale, real-world applications. For geometric multigrid, it is known that using p-coarsening before h-coarsening can provide better scalability, but setting up p-coarsening is non-trivial in AMG. We develop a geometrically informed algebraic multigrid (GIAMG) method, as well as an associated high-performance computing program, which is able to set up a grid hierarchy that includes p-coarsening at the top grids with minimal information of the geometry from the user. A major advantage of using p-coarsening with AMG -- beyond the benefits known in the context of geometric multigrid (GMG) -- is the increased sparsification of coarse grid operators. We extensively evaluate GIAMG by testing on the 3D Helmholtz and incompressible flow problems, and demonstrate mesh-independent convergence, and excellent parallel scalability. We also compare the performance of GIAMG with existing AMG packages, including Hypre and ML.