General description of multigrid algorithms

Abstract

In section 4.1, we deal with the auxiliary results of solving the problems of iterative parameters optimization. In four following sections, the description of multigrid algorithms is performed at an abstract algebraic level. It gives the possibility to utilize the main results both to finite-element schemes and finite-difference ones. Here we use designations and approaches closed to works [31], [68]. For example, in these works, the abstract algebraic results are transformed into terms of finite-difference schemes for the second-order elliptic equations. In section 4.2, the basic multigrid algorithm for suppression of error of approximate solution is formulated and the full nested algorithm is described on the sequence of ascending levels. In section 4.3, we study the convergence of multigrid algorithms (to be more precise, multilevel) for problems with a self-ajoint positive definite operator. First, the convergence is stated for an arbitrary smoother and then the description is elaborated for generalized Jacoby-type smoothing processes. The point is that we succeed in finding the optimal iterative parameters so that the multiplier of error suppression in various norms is better (with respect to number m of iterations) than in the large majority of iterative smoothers. One can see this advance comparing various smoothers in monograph [68]. Even if the convergence rate is the same with some other iterative procedure, this smoother is usually simpler in computations. For example, one may use only elementary matrices of stiffness and mass without their storage and assembling.