Toward a multilevel method for the Helmholtz equation
Résumé
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, two main issues may arise when solving a Helmholtz problem. Some eigenvalues become negative or even complex, requiring the choice of an adapted smoothing method for capturing them. Moreover, since the near-kernel space is oscillatory, the geometric smoothness assumption cannot be used to build efficient interpolation rules. 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 correct an initial least squares minimization coarse selection operator formed from a set of smoothed random vectors. We also present numerical results at the end of the paper.
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