Analysis of two-grid methods: The nonnormal case

Core results about the algebraic analysis of two-grid methods are extended in relations bounding the field of values (or numerical range) of the iteration matrix. On this basis, bounds are obtained on its norm and numerical radius, leading to rigorous convergence estimates. Numerical illustrations show that the theoretical results deliver qualitatively good predictions, allowing one to anticipate success or failure of the two-grid method. They also indicate that the field of values and the associated numerical radius are much more reliable convergence indicators than the eigenvalue distribution and the associated spectral radius. On this basis, some discussion is developed about the role of local Fourier or local mode analysis for nonsymmetric problems.