The $V$-cycle multigrid convergence of some finite difference scheme for the Helmholtz equation

In this paper, we analyze the $V$-cycle multigrid algorithm for a positive definite Helmholtz equation on a hexagonal grid. Specifically, we apply the $V$-cycle multigrid algorithm to the numerical scheme based on the mean value solutions for the Helmholtz equation on hexagonal grids introduced in [1], and show its convergence. The theory for the $V$-cycle multigrid convergence is carried out in the framework in [6] by estimating the energy norm of the prolongation operator and proving the approximation and regularity conditions. In numerical experiments, we report the eigenvalues, condition number and contraction number.