Abstract
In this paper, using the quadratic spline collocation method (QSC), we numerically solve the Helmholtz equations with the Sommerfeld boundary conditions. By reordering the unknowns, we obtain a $$3\times 3$$ block linear system. Then, we introduce a two-step preconditioner based on the approximate inverse block polynomial preconditioner. Theoretical analysis show this preconditioner can largely gather the eigenvalues around 1. Numerical examples are presented to test the error of QSC method and check the efficiency of the presented preconditioner.
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