A SCALABLE HELMHOLTZ SOLVER COMBINING THE SHIFTED LAPLACE PRECONDITIONER WITH MULTIGRID DEFLATION

A Helmholtz solver whose convergence is parameter independent can be obtained by combining the shifted Laplace preconditioner with multigrid deflation. To proof this claim, we develop a Fourier analysis of a two-level variant of the algorithm proposed in [1]. In this algorithm those eigenvalues that prevent the shifted Laplace preconditioner from being scalable are removed by deflation using multigrid vectors. Our analysis shows that the spectrum of the two-grid operator consists of a cluster surrounded by a few outliers, yielding a number of outer Krylov subspace iterations that remains constant as the wave number increases. Our analysis furthermore shows that the imaginary part of the shift in the two-grid operator can be made arbitrarily large without affecting the convergence. This opens promising perspectives on obtaining a very good preconditioner at very low cost. Numerical tests for problems with constant and non-constant wave number illustrate our convergence theory.

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