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Toward a multilevel method for the Helmholtz equation


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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hal-04046622 , version 1 (27-03-2023)


  • HAL Id : hal-04046622 , version 1


Clément Richefort, Matthieu Lecouvez, Robert Falgout, Pierre Ramet. Toward a multilevel method for the Helmholtz equation. 21st SIAM Copper Mountain Conference on Multigrid Method, Apr 2023, Copper Mountain, CO, United States. ⟨hal-04046622⟩
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