Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers

We revisit the Hierarchical Poincar\'e--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.