Revistin Multigrid for the Helmholtz Problem

It is well known that standard multigrid algorithms are ineffective to solve the discrete Helmholtz problems and there are several well-written articles that try to interpret this difficulty.


In this presentation, we reconsider the multigrid for the discretizations of the Helmholtz problem and bring new insights. We propose a new simple Fourier analysis (SFA) which is different from the existing one in literature, and very easy to understand. With SFA, we obtain all spectral information of the iteration matrix of the two-grid for a generic symmetric tridiagonal matrix. Then, we examine multigrid for the Helmholtz problem in 1D as its discretizations with Dirichlet boundary condition, give symmetric tridiagonal matrices. Our analyses show that there are fundamental issues in the interpretation of the multigrid for the Helmholtz problem. Moreover, we reinterpret the problematic coarse levels for the multigrid and bring a restriction on the coarse levels. This restriction reduces the computational cost for the existing multigrid algorithms, significantly. Based on this observation, we propose a convergent multigrid algorithm.