Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers

TL;DR

The paper addresses efficient solution of variable-coefficient Helmholtz problems by reframing the Hierarchical Poincaré–Steklov method as a multilevel preconditioner. It unifies direct and iterative perspectives by treating the HPS hierarchical merges as Schur-complementbased coarse operators, enabling a recursive multigrid relaxation that preserves spectral discretization accuracy. The approach yields a flexible, scalable preconditioner for GMRES and demonstrates 2D convergence behavior with tunable coarse-cycle parameters on large problems. This framework has potential practical impact for large-scale wave propagation simulations in heterogeneous media, offering tunable memory and compute trade-offs while maintaining high-order accuracy.

Abstract

We revisit the Hierarchical Poincaré--Steklov (HPS) method within a preconditioned iterative framework. Originally introduced as a direct solver for elliptic boundary-value problems, the HPS method combines nested dissection with tensor-product spectral element discretizations, even though it has been shown in other contexts[8]. Building on the iterative variant proposed in[1], we reinterpret the hierarchical merge structure of HPS as a natural multigrid preconditioner. This perspective unifies direct and iterative formulations of HPS connecting it to multigrid domain decomposition. The resulting formulation preserves the high accuracy of spectral discretizations while enabling flexible iterative solution strategies. Numerical experiments in two dimensions demonstrate the performance and convergence behavior of the proposed approach.

Figures (2)

• Figure 1: Face merging and sparsity patterns for a $4\times 4$ element mesh
• Figure 2: Solution of the variable-coefficient Helmholtz problem.