A basic error bound for multigrid methods is given in terms of residuals on neighboring levels. The terms in this bound derive from the iterative methods used as solvers on each level and the operators used to go from a level to the next coarser level. This bound is correct whether the underlying operator is symmetric or nonsymmetric, definite or indefinite, and singular or nonsingular. We allow any iterative method as a smoother (or rougher) in the multigrid cycle.
One of the advantages of this theory is that all of the parameters are available during execution of a computer program. Hence, adaptively changing levels can be achieved with certainty of success. This is particularly important for solving problems in which there is no known useful convergence analysis. Two problems arising in modeling combustion problems (flame sheets and laminar diffusion flames with full chemistry) are discussed.
While this theory is quite general, it is not always the correct approach when analyzing the convergence rate for a given problem. A discussion of when this theory is useful and when it is hopelessly nonsharp is provided.