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

Authors: I. G. Graham and S. A. Sauter
Journal: Math. Comp. 89 (2020), 105-138
MSC (2010): Primary 35J05, 65N12, 65N15, 65N30
Published electronically: July 1, 2019
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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 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.

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Additional Information

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

S. A. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Keywords: Helmholtz equation, high frequency, variable wave speed, variable density, well-posedness, a priori estimates, {finite element error analysis}
Received by editor(s): March 2, 2018
Received by editor(s) in revised form: October 11, 2018, and February 3, 2019
Published electronically: July 1, 2019
Article copyright: © Copyright 2019 American Mathematical Society