A two-level analysis method for certain separable problems is introduced. Unlike standard two-level analysis methods, based on Fourier analysis, it is based on spectral analysis, hence applicable to non-normal problems and to certain problems with variable coefficients. It motivates the definition of improved versions of Black Box Multigrid for diffusion problems with discontinuous coefficients and indefinite Helmholtz equations. For anisotropic problems, it helps in choosing suitable implementations for frequency decomposition multigrid methods. For highly indefinite problems, it provides a way to choose a suitable mesh size for the coarsest grid used. Numerical experiments confirm the analysis and show the advantage of the present methods.