On the Optimality of Shifted Laplacian in a Class of Polynomial Preconditioners for the Helmholtz Equation
This paper introduces and explores a class of polynomial preconditioners for the Helmholtz equation, denoted as expansion preconditioners EX(m), that form a direct generalization to the classical complex shifted Laplace (CSL) preconditioner. The construction of the EX(m) preconditioner is based on a truncated Taylor series expansion of the original Helmholtz operator inverse. The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for growing values of the number of series terms m. However, the addition of multiple terms in the expansion also increases the computational cost of applying the preconditioner. A thorough cost-benefit analysis of the addition of extra terms in the EX(m) preconditioner proves that the CSL or EX(1) preconditioner is the most efficient member of the expansion preconditioner class for general practical and solver problem settings. Additionally, possible extensions to the expansion preconditioner class that further increase preconditioner efficiency are suggested, and numerical experiments in 1D and 2D are presented to validate the theoretical results.
KeywordsHelmholtz equation Krylov subspace methods Multigrid methods Preconditioning Shifted Laplacian
This research in this chapter was funded by the Research Council of the University of Antwerp under BOF grant number 41-FA070300-FFB5342. The scientific responsibility rests with its authors. The authors would like to cordially thank Dr. Bram Reps for fruitful discussions on the subject.
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