Stability and finite element error analysis for the Helmholtz equation with variable coefficients

We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly nonsmooth 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 a 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 the stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means of a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp