Algebraic multigrid for k-form Laplacians

In this paper we describe an aggregation-based algebraic multigrid method for the solution of discrete k-form Laplacians. Our work generalizes Reitzinger and Schöberl's algorithm to higher-dimensional discrete forms. We provide conditions on the tentative prolongators under which the commutativity of the coarse and fine de Rham complexes is maintained. Further, a practical algorithm that satisfies these conditions is outlined, and smoothed prolongation operators and the associated finite element spaces are highlighted. Numerical evidence of the efficiency and generality of the proposed method is presented in the context of discrete Hodge decompositions.