Stability and error analysis for the Helmholtz equation with variable coefficients

Ivan Graham, Stefan Sauter

Research output: Contribution to journalArticlepeer-review

Abstract

We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e. PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz equation which is valid at high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp
Original languageEnglish
Pages (from-to)105-138
Number of pages24
JournalMathematics of Computation
Volume89
Issue number321
Early online date1 Jul 2019
DOIs
Publication statusPublished - 2020

Fingerprint Dive into the research topics of 'Stability and error analysis for the Helmholtz equation with variable coefficients'. Together they form a unique fingerprint.

Cite this