Getting standard multigrid to work efficiently for the high-frequency Helmholtz equation has been an open problem in applied mathematics for years. Much effort has been dedicated to finding solution methods which can use multigrid components to obtain solvers with a linear time complexity. In this work we present one among the first stand-alone multigrid solvers for the 2D Helmholtz equation using both a constant and non-constant wavenumber model problem. We use standard smoothing techniques and do not impose any restrictions on the number of grid points per wavelength on the coarse-grid. As a result we are able to obtain a full V- and W-cycle algorithm. The key features of the algorithm are the use of higher-order inter-grid transfer operators combined with a complex constant in the coarsening process. Using weighted-Jacobi smoothing, we obtain a solver which is $h-$independent and scales linearly with the wavenumber $k$. Numerical results using 1 to 5 GMRES(3) smoothing steps approach $k-$ and $h-$ independent convergence, when combined with the higher-order inter-grid transfer operators and a small or even zero complex shift. The proposed algorithm provides an important step towards the perpetuating branch of research in finding scalable solvers for challenging wave propagation problems.

翻译：如何使标准多重网格高效求解高频亥姆霍兹方程，一直是应用数学领域多年未解的难题。大量研究致力于寻找能够利用多重网格组件实现线性时间复杂度的求解方法。本文提出了首批适用于二维亥姆霍兹方程的独立多重网格求解器之一，涵盖常波数与变波数模型问题。我们采用标准平滑技术，且不对粗网格每波长网格点数施加任何限制，从而实现了完整的V循环与W循环算法。该算法的关键特征在于：粗化过程中采用高阶网格间传递算子，并结合复常数。通过加权雅可比平滑，我们获得了与网格步长$h$无关、且随波数$k$线性扩展的求解器。数值实验表明：当采用1至5步GMRES(3)平滑、结合高阶网格间传递算子与微小甚至零复移项时，收敛性趋近于与$k$和$h$无关。所提算法为求解具有挑战性的波传播问题、寻找可扩展求解器的持续性研究提供了重要进展。