Abstract
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 existenceuniqueness 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 language  English 

Pages (fromto)  105138 
Number of pages  24 
Journal  Mathematics of Computation 
Volume  89 
Issue number  321 
Early online date  1 Jul 2019 
DOIs  
Publication status  Published  2020 
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Profiles

Ivan Graham
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching