Generalized grid transfer operators for multigrid methods applied on Toeplitz matrices

In this paper we discuss classical sufficient conditions to be satisfied from the grid transfer operators in order to obtain optimal two-grid and V-cycle multigrid methods utilizing the theory for Toeplitz matrices. We derive relaxed conditions that allow the construction of special grid transfer operators that are computationally less expensive while preserving optimality. This is particularly useful when the generating symbol of the system matrix has a zero of higher order, like in the case of higher order PDEs. These newly derived conditions allow the use of rank deficient grid transfer operators. In this case the use of a pre-relaxation iteration that is lacking the smoothing property is proposed. Combining these pre-relaxations with the new rank deficient grid transfer operators yields a substantial reduction of the convergence rate and of the computational cost at each iteration compared with the classical choice for Toeplitz matrices. The proposed strategy, i.e. a rank deficient grid transfer operator plus a specific pre-relaxation, is applied to linear systems whose system matrix is a Toeplitz matrix where the generating symbol is a high-order polynomial. The necessity of using high-order polynomials as generating symbols for the grid transfer operators usually destroys the Toeplitz structure on the coarser levels. Therefore, we discuss some effective and computational cheap coarsening strategies found in the literature. In particular, we present numerical results showing near-optimal behavior while keeping the Toeplitz structure on the coarser levels.