Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Summary.

The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13,14,22]. The technique was modified in [6,36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at $x^0=0$ of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at $x^0=0$ of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed.