two-level method; pre-smoothing; coarse-grid correction; post-smoothing; error propagation; non-scalar elliptic problems; system of linear algebraic equations; symmetric positive definite matrix; smoothing; algorithms; aggregation method
We derive the smoothed aggregation two-level method from the variational objective to minimize the final error after finishing the entire iteration. This contrasts to a standard variational two-level method, where the coarse-grid correction vector is chosen to minimize the error after coarse-grid correction procedure, which represents merely an intermediate stage of computing. Thus, we enforce the global minimization of the error. The method with smoothed prolongator is thus interpreted as a qualitatively different, and more optimal, algorithm than the standard multigrid.