Remarks on multigrid convergence theorems

摘要

Multigrid has become an important iterative method for the solution of discrete elliptic equations. However, there is much to be done in the theory of convergence proofs. At the present time there are two general two-level methods for general convergence proofs: an algebraic approach and a duality approach. While these theories do not give sharp estimates, they provide good, general, rigorous convergence theorems. In this note we study the relationship between these theories. While the approach and thought process leading to these theories are different, the results are essentially the same. Indeed, the basic estimates required by these theories are the same.