Fourier Analysis of GMRES(m) Preconditioned by Multigrid

This paper deals with convergence estimates of GMRES(m) [Saad and Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869] preconditioned by multigrid [Brandt, Math. Comp., 31 (1977), pp. 333–390], [Hackbusch, Multi-Grid Methods and Applications, Springer, Berlin, 1985]. Fourier analysis is a well-known and useful tool in the multigrid community for the prediction of two-grid convergence rates [Brandt, Math. Comp., 31 (1977), pp. 333–390], [Stüben and Trottenberg, in Multigrid Methods, Lecture Notes in Math. 960, K. Stüben and U. Trottenberg, eds., Springer, Berlin, pp. 1–176]. This analysis is generalized here to the situation in which multigrid is a preconditioner, since it is possible to obtain the whole spectrum of the two-grid iteration matrix. A preconditioned Krylov subspace acceleration method like GMRES(m) implicitly builds up a minimal residual polynomial. The determination of the polynomial coefficients is easily possible and can be done explicitly since, from Fourier analysis, a simple block-diagonal two-grid iteration matrix results. Based on the GMRES(m) polynomial, sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix. Several numerical scalar test problems are computed in order to validate the theoretical predictions.