Abstract:

In this paper, we apply Fourier analysis (including local Fourier analysis and general Fourier analysis) of multi-grid method to predict and analyze the convergence of multi-grid method used to discretize Helmholtz equations with complex-valued entries aroused in magnetotelluric problem. The Fourier spectra, which are usually complex in spectral domain,however, should be converted to real domain when visualizing the convergence behavior. The eigenvector spectra in real domain could be used to answer the slow convergence in two-grid method with Gauss-Seidel solver used on the coarsest grid. As not including boundary conditions and variant coefficients, local Fourier analysis (e.g., one-grid Fourier analysis and two-grid Fourier analysis) is difficult to give a reasonable asymptotic convergence estimate of two-grid method with a direct solver on the coarsest grid, whereas two-grid general Fourier analysis could. The Fourier analysis for five-grid method using a direct solver on coarsest grid shows that when the general higher-grid Fourier analysis is applied, the general Fourier analysis gets close to the numerical asymptotic convergence. Based on the set of general multi-grid Fourier analysis results, an empirical formula approximating the asymptotic convergence for five-grid method is derived,thus we just need to carry out the general low-grid Fourier analysis(two-grid and three-grid) on coarsest grids and approximate the asymptotic convergence with low cost.

Key words: multigrid method, Fourier analysis, coarsest grid, convergence, magnetotellurics