Optimal polynomial smoothers for multigrid V‐cycles

Abstract

AbstractThe idea of using polynomial methods to improve simple smoother iterations within a multigrid method for a symmetric positive definite system is revisited. A two‐level bound going back to Hackbusch is optimized by a very simple iteration, a close cousin of the Chebyshev semi‐iterative method, but based on the Chebyshev polynomials of the fourth instead of first kind. A full V‐cycle bound for general polynomial smoothers is derived using the V‐cycle theory of McCormick. The fourth‐kind Chebyshev iteration is quasi‐optimal for the V‐cycle bound. The optimal polynomials for the V‐cycle bound can be found numerically, achieving about an 18% lower error contraction factor bound than the fourth‐kind Chebyshev iteration, asymptotically as the number of smoothing steps . Implementation of the optimized iteration is discussed, and the performance of the polynomial smoothers is illustrated with numerical examples.