  This paper introduces a novel approach to algebraic multigrid methods for
large systems of linear equations coming from finite element discretizations of
certain elliptic second order partial differential equations. Based on a
discrete energy made up of edge and vertex contributions, we are able to
develop coarsening criteria that guarantee two-level convergence even for
systems of equations. This energy also allows us to construct prolongations
with prescribed sparsity pattern that still preserve kernel vectors exactly.
These allow for a straightforward optimization that simplifies parallelization
and reduces communication on coarse levels. Numerical experiments demonstrate
efficiency and robustness of the method and scalability of the implementation.
