.It is well-known that multigrid methods are very competitive in solving a wide range of SPD problems. However, achieving such performance for non-SPD matrices remains an open problem. In particular, three main issues may arise when solving a Helmholtz problem: some eigenvalues may be negative or even complex, requiring the choice of an adapted smoother for capturing them, and because the near-kernel space is oscillatory, the geometric smoothness assumption cannot be used to build efficient interpolation rules. Moreover, the coarse correction is not equivalent to a projection method since the indefinite matrix does not define a norm. We present some investigations about designing a method that converges in a constant number of iterations with respect to the wavenumber. The method builds on an ideal reduction-based framework and related theory for SPD matrices to improve an initial least squares minimization coarse selection operator formed from a set of smoothed random vectors. A new coarse correction is proposed to minimize the residual in an appropriate norm for indefinite problems. We also present numerical results at the end of the paper.Keywordsalgebraic multigridHelmholtz equationlinear algebraindefinite matrixMSC codes65F10
Falgout et al. (Thu,) studied this question.