Sharp Estimates for Multigrid Rates of Convergence with General Smoothing and Acceleration

Abstract

Abstract. In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. Our theory is presented entirely in terms of the generalized condition number K of the matrix A and the smoothing matrix B. This leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations; the properties of the elliptic equation and discretization procedure enter only when we seek to estimate K, just as in the case of most standard iterative methods. Here we consider the fundamental two-level iteration, and the V and W cycles of the j-level iteration (j> 2). We prove that the V and W cycles converge even when only one smoothing iteration is used. We present several examples of the computation of using both Fourier analysis and standard finite element techniques. We compare the predictions of our theorems with the actual rate of convergence. Our analysis also shows that accelerated iterative methods, both fixed (Chebyshev) and adaptive (conjugate gradients and conjugate residuals), are effective as smoothing procedures.

The analysis of smoothers for multigrid algorithms

“…In conjunction, there has been intensive research into the theoretical understanding of these methods (cf. [1,2,3,4,6,9,11,12,13,14,18] and others). Many of the above papers present various analyses of multigrid methods which are often based on certain assumptions concerning the smoothing process.…”

The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms

“…Because of this variable smoothing, the above algorithm is more general than that usually described [2,3,8,16]. If Rk is symmetric and all of the m(k) are the same, then the algorithm is the usual symmetric multigrid cycling algorithm described in a notation which is convenient for our analysis.…”

“…We note that many of the results given in this paper (and also [2,4,5,6,20] in the variational case) provide multigrid analysis for any number of smoothings on the finest grid. Such results are important to the code developer in that they guarantee that algorithms will work with just one smoothing.…”

“…In conjunction, there has been intensive research into the theoretical understanding of these methods (cf. [2,3,4,5,6,16,20,21,22,30] and others). In this paper, we shall extend the theory for symmetric problems so that it applies in a more general framework.…”

Algebraic Theory of Two-Grid Methods

“…In particular, one should not forget that more general or sharper bounds are always indebted to their predecessors: even when the new proof is very different in nature, it has in general been guided by the intuition of the result that one aims at. Besides the works cited later in this paper, it is thus also worth mentioning seminal papers like [3,5,20,26,31,62] that occupy a middle ground between algebraic and non algebraic convergence theories.…”