Characterization of the near-null error components utilized in composite adaptive AMG solvers

We provide a theoretical justification for the construction of adaptive composite solvers based on a sequence of AMG (algebraic multigrid) $\mu$-cycle methods that exploit error components that the current solver cannot damp efficiently. Each solver component is an aggregation based AMG where its aggregates are constructed using the popular in graph community detection modularity matrix. The latter utilizes the given matrix and the error component vector the current solver cannot handle. The performance of the resulting adaptive composite solver is illustrated on a variety of sparse matrices both arising from discretized PDEs and ones with more general nature.