Geodesic
The $V$-cycle multigrid convergence of some finite difference scheme for the Helmholtz equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 3, pp. 207-218

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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. 
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     author = {Kwak Do Y. and Lee Jun S.},
     title = {The $V$-cycle multigrid convergence of some finite difference scheme for the {Helmholtz} equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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Kwak Do Y.; Lee Jun S. The $V$-cycle multigrid convergence of some finite difference scheme for the Helmholtz equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 3, pp. 207-218. http://geodesic.mathdoc.fr/item/SJVM_2005_8_3_a2/