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

TL;DR

The paper addresses the solver challenges of high-order finite element methods by introducing GIAMG, a geometrically informed algebraic multigrid preconditioner that uses minimal geometric information to perform $p$-coarsening at the top levels before applying $h$-coarsening. The approach yields sparser coarse-grid operators and improved scalability, demonstrated through 3D Helmholtz and incompressible-flow tests where GIAMG shows mesh-independent convergence and strong parallel performance, outperforming several established AMG packages. Detailed analyses reveal the impact of smoothing iterations, $p$-coarsening strategies, and strong scaling up to 128 nodes. The authors provide open-source code and outline future work to automate geometry extraction from matrices and extend applicability to non-symmetric problems.

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.

Figures (11)

• Figure 1: Illustration of the coarsening strategy of GIAMG for quadrilateral elements with nodal basis of polynomial order $p=4$. First, we $p$-coarsen from $p=4$ to $p=2$, and then again $p$-coarsen from $p=2$ to $p=1$. The total number of elements---and therefore the mesh---does not change during $p$-coarsening. Once we reach linear elements, we perform $h$-coarsening to obtain a coarser mesh and smaller system.
• Figure 2: Convergence results of GIAMG from $p$ = 3 to 10 for the Helmholtz operator.
• Figure 3: Convergence and solve time comparisons of GIAMG with PETSc GAMG, ML, BoomerAMG and DCG at $p$ = 8 for the Helmholtz operator.
• Figure 4: Geometry and mesh of the computational domain.
• Figure 5: Convergence results for $p$ from 1 to 5 for the incompressible flow problem.