Improved convergence bounds for two-level methods with an aggressive coarsening and massive polynomial smoothing. ETNA - Electronic Transactions on Numerical Analysis

An improved convergence bound for the polynomially accelerated two-level method of Brousek et al. [Electron. Trans. Numer. Anal., 44 (2015), pp. 401–442, Section 5] is proven. This method is a reinterpretation of the smoothed aggregation method with an aggressive coarsening and massive polynomial smoothing of Vaněk, Brezina, and Tezaur [SIAM J. Sci. Comput., 21 (1999), pp. 900–923], and its convergence rate estimate is improved here quantitatively. Next, since the symmetrization of the method requires two solutions of the coarse problem, a modification of the method is proposed that does not have this disadvantage, and a qualitatively better convergence result for the modification is established. In particular, it is shown that a bound of the convergence rate of the method with a multiply (k-times) smoothed prolongator is asymptotically inversely proportional to d^{2k}, where d is the degree of the smoothing polynomial. In earlier works, this acceleration effect is only quadratic. Finally, for another modified multiply smoothed method, it is proved that this convergence improvement is not limited only to an asymptotic regime but holds true everywhere