Multigrid methods are fast iterative solvers for partial di erential equations. Especially for elliptic equations they have been proven to be highly e cient. For problems with nonelliptic and nonsymmetric features--as they often occur in typical real life applications--a rigorous mathematical theory is generally not available. For such situations Fourier smoothing and two-grid analysis can be considered as the main analysis tools to obtain quantitative convergence estimates and to optimize different multigrid components like smoothers or inter-grid transfer operators. In general, it is difficult to choose the correct multigrid components for large classes of problems. A popular alternative to construct a robust solver is the use of multigrid as a preconditioner for a Krylov subspace acceleration method like GMRES. Our contributions to the Fourier analysis for multigrid are two-fold. Firstly we extend the range of situations for which the Fourier analysis can be applied. More precisely, the Fourier analysis is generalized to k-grid cycles and to multigrid as a preconditioner. With a k-grid analysis it is possible to investigate real multigrid effects which cannot be captured by the classical two-grid analysis. Moreover, the k-grid analysis allows for a more detailed investigation of possible coarse grid correction difficulties. Additional valuable insight is obtained by evaluating multigrid as a preconditioner for GMRES. Secondly we extend the range of discretizations and multigrid components for which detailed Fourier analysis results exist. We consider four well-known singularly perturbed model problems to demonstrate the usefulness of the above generalizations: The anisotropic Poisson equation, the rotated anisotropic diffusion equation, the convection diffusion equation with dominant convection, and the driven cavity problem governed by the incompressible Navier Stokes equations. Each of these equations represents a larger class of problems with similar features and complications which are of practical relevance. With the help of the newly developed Fourier analysis methods, a comprehensive study of characteristic difficulties for singular perturbation problems can be performed. Based on the insights from this analysis it is possible to identify remedies resulting in an improved multigrid convergence. The theoretical considerations are validated by numerical test calculations
