Coarse Spaces Based on Higher-Order Interpolation for Schwarz Preconditioners for Helmholtz Problems

TL;DR

The paper tackles the difficulty of obtaining scalable, wavenumber-robust solvers for the Helmholtz equation by enhancing two-level Schwarz domain decomposition preconditioners with coarse spaces built from higher-order Bézier interpolation. The coarse space is constructed on a coarse grid using ${\mathbb{B}_2}^H$ interpo lation, and several two-level variants (AS2, SAS2, SHS2) are analyzed, with SHS2 incorporating a deflation projection to suppress low-frequency modes. Numerical experiments on MP-1 and MP-2 demonstrate that the higher-order Bézier coarse space yields robust convergence as the wavenumber increases, especially for the SHS2 preconditioner, while linear coarse spaces degrade in performance. The findings indicate that wavenumber robustness requires a sufficiently refined coarse mesh (e.g., $kH \le 1$), and the proposed approach offers a practical, scalable path toward robust Helmholtz solvers for large-scale problems.

Abstract

The development of scalable and wavenumber-robust iterative solvers for Helmholtz problems is challenging but also relevant for various application fields. In this work, two-level Schwarz domain decomposition preconditioners are enhanced by coarse space constructed using higher-order Bézier interpolation. The numerical results indicate numerical scalability and robustness with respect the wavenumber, as long as the wavenumber times the element size of the coarse mesh is sufficiently low.

Figures (2)

• Figure 1: Analytical solution of MP-1 with $k = 50$ on the unit square domain with homogeneous Dirichlet boundary conditions and the point source term located at the centre of the domain.
• Figure 2: Higher-order Bézier interpolation function for a fine and coarse grid with $H = 2h$.