Adaptive aggregation on graphs

We generalize some of the functional (hypercircle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. Several theoretical results in regard to the generalized a posteriori error estimators are provided. We use these estimates to construct aggregation based coarse spaces for graph Laplacians. The estimator is used to assess the quality of an aggregation adaptively. Furthermore, a reshaping based algorithm is tested on several numerical examples.