Convergence estimates for multigrid algorithms

Abstract

To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the multigrid literature [1,2]. However, numerical evidence shows that Kaczmarz smoother does not satisfy above assumptions. Thus, we introduce a weaker condition which is satisfied by Kaczmarz smoother as well as Jacobi and Gauss-Seidel smoother. Under these weaker assumptions, we show that the convergence factor of V-cycle multigrid algorithm is delta = 1-1/(C(j-1)). These assumptions for Kaczmarz smoother are verified by numerical experiment.