Convergence Analysis of an Aggregation‐Based Two‐Grid Method, With Applications to Linear Elasticity

We present a convergence analysis of a Galerkin‐type two‐grid method based on plain (or unsmoothed) aggregation. The analysis assumes that the system matrix is symmetric nonnegative definite (SNND) and can be decomposed into (or bounded below by) a sum of SNND matrices, each associated with an individual aggregate. Then, the convergence factor of the considered two‐grid method can be bounded above (up to the smoother's scaling) as a function of a maximum over all aggregates of a quantity associated to each aggregate and named aggregate's quality. The presented analysis is a generalization of the analysis from [A. Napov and Y. Notay, Numer. Linear Algebra Appl. 18(2011), pp. 539–564], and is suitable for finite element discretizations, for which the system matrix subdivision into element matrices is a natural starting point. The analysis is further applied in the context of standard finite element discretizations of linear elasticity problems. More specifically, we derive a sharp estimate of the quality parameter of a two‐node aggregate, that is, an aggregate containing all unknowns associated to two connected nodes of the discretization mesh. This estimate is given here as a function of the average stiffness tensor (which we do not assume isotropic), the vector connecting the nodes, the geometry of the elements containing both nodes and the diagonal entries of the system matrix corresponding to the nodes. Eventually, we illustrate the accuracy and usefulness of the presented bounds with some numerical experiments.