A multigrid-based preconditioner for solving the Helmholtz equation with average-derivative optimal scheme

Abstract

An efficient finite-difference method for solving Helmholtz equation depends on two points: one is discrete scheme, the other is efficient algorithm. In this paper, we adopt the average-derivative scheme, which owns three advantages : Firstly, it can be applied to unequal directional sampling intervals for Helmholtz equation. Secondly, the scheme is pointwise consistent with Helmholtz equation in a perfect matched layer. And thirdly, it requires less than 4 grid points sampling per wavelength. To solve the discrete Helmholtz equation, which is extremely large and indefinite, direct methods cannot resolve well, and the Krylov subspace iterative methods, such as Bi-CGSTAB and GMRES combining a multigrid-based preconditioner, are good choices. However, the standard multigrid algorithm fails to converge when it encounters unequal directional sampling intervals, which is called anisotropy in multigrid. We analyze the most important three parts of standard multigrid : full weighting restriction operator, point relaxation methods and bilinear interpolation operator, and then we replace them with semi-coarsening, line relaxation and operator-dependent interpolation to make it convergent in anisotropic problems. Consequently, we obtain a satisfactory convergence speed for low and moderate frequency iterative problems in heterogeneous media.