Energy optimization of algebraic multigrid bases

We propose a fast iterative method to optimize coarse basis functions in al gebraic multigrid by minimizing the sum of their energies, subject to the c ondition that linear combinations of the basis functions equal to given zer o energy modes, and subject to restrictions on the supports of the coarse b asis functions. For a particular selection of the supports, the first itera tion gives exactly the same basis functions as our earlier method using smo othed aggregation. The convergence rate of the minimization algorithm is bo unded independently of the mesh size under usual assumptions on finite elem ents. The construction is presented for scalar problems as well as for line ar elasticity. Computational results on difficult industrial problems demon strate that the use of energy minimal basis functions improves algebraic mu ltigrid performance and yields a more robust multigrid algorithm than smoot hed aggregation.