Abstract
In this paper we analyze a preconditioning technique for the solution of Chebyshev spectral collocation equations with Dirichlet boundary conditions. We obtain bounds on the eigenvalues for the Helmholtz equation. These eigenvalue bounds are obtained as a consequence of estimates on the field of values $(\tilde A_{N^2}U,U)_{l_2}/(\tilde Q_{N^2}U,U)_{l_2}$, where $\tilde A_{N^2}$ is the weighted collocation matrix and $\tilde Q_{N^2}$ is the preconditioner. The preconditioner $\tilde Q_{N^2}$ is robust in the sense that it provides bounds on the $H^1_{0,w}$ condition number of $\tilde Q_{N^2}^{-1}\tilde L_{N^2}$ when $\tilde L_{N^2}$ is the weighted collocation matrix associated with the general elliptic operator $Lu:= -\Delta u + a_1u_x + a_2u_y + a_0u$.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
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.