On an energy minimizing basis for algebraic multigrid methods
Energy minimal basis functions (EMBF) result in good multigrid convergence. EMBF and a numerical method for computing EMFB were first proposed by \textit{W. L. Wan, T. F. Chan}, and \textit{B. Smith} [SIAM J. Sci. Comput. 21, 1632--1649 (2000; Zbl 0966.65098)] but the numerical method is expensive. Smoothed aggregation constructs approximate EMBF, and a fast primal iterative method of projected gradient descent type to compute EMBF is obtained by more smoothing [cf. \textit{J. Mandel, M. Brezina}, and \textit{P. Vanek}, Computing 62, 205--228 (1999; Zbl 0942.65034)]. In this paper, a fast dual iterative method is proposed to compute EMBF. The new method is based on the inversion of an operator of abstract additive Schwarz type, which expresses the optimality condition of the energy in local subspaces. This leads to a well conditioned problem for the Lagrange multipliers. It is shown that the Lagrange multipliers are edge discrete harmonic and that EMBF are discrete harmonic on aggregated elements. There are numerical examples.
scientific article; zbMATH DE number 2175873
scientific article