Abstract
We introduce a new polynomial preconditioner for solving the discretized Helmholtz equation preconditioned with the complex shifted Laplace (CSL) operator. We exploit the localization of the spectrum of the CSL-preconditioned system to approximately enclose the eigenvalues by a non-convex âbratwurstâ set. On this set, we expand the function 1/z into a Faber series. Truncating the series gives a polynomial, which we apply to the Helmholtz matrix preconditioned by the shifted Laplacian to obtain a new preconditioner, the Faber preconditioner. We prove that the Faber preconditioner is nonsingular for degrees one and two of the truncated series. Our numerical experiments (for problems with constant and varying wavenumber) show that the Faber preconditioner reduces the number of GMRES iterations.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: ETNA - Electronic Transactions on Numerical Analysis
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
