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On an energy minimizing basis for algebraic multigrid methods. (English) Zbl 1077.65130
Energy minimal basis functions (EMBF) result in good multigrid convergence. EMBF and a numerical method for computing EMFB were first proposed by W. L. Wan, T. F. Chan, and 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. J. Mandel, M. Brezina, and 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.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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