Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincar\'e-Steklov Solvers

Abstract

We revisit the Hierarchical Poincar\'e-Steklov (HPS) method in a preconditioned iterative setting for variable-coefficient Helmholtz problems with impedance boundary conditions. HPS is commonly presented as a direct solver based on nested dissection and high-order tensor-product discretizations; here we recast its hierarchical merge tree as a multilevel preconditioner for the assembled skeleton (trace) system. The main goal is to flexibilize the final, memory-intensive coarse stage of direct HPS by replacing the exact coarse solve with a small, fixed amount of iterative work, thereby exposing tunable trade-offs between memory footprint and time to solution. Numerical experiments on a two-dimensional scattering benchmark illustrate these trade-offs and compare against both unpreconditioned GMRES and the classic direct HPS pipeline with an exact coarse space.