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IMA Journal of Applied Mathematics 1999 63(1):51-69; doi:10.1093/imamat/63.1.51
© 1999 by Institute of Mathematics and its Applications
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Continuous Fréchet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

R Djellouli, C Farhat, J MandelY and P VanêkY

Department of Aerospace Engineering Sciences, and Center for Aerospace Structures, University of Colorado at Boulder, Boulder, Colorado 80309-0429, USA Y Also at: Department of Mathematics, University of Colorado at Denver, Denver, CO 80217-3364, USA

We consider direct acoustic scattering problems with either a sound-soft or sound-hard obstacle, or lossy boundary conditions, and establish continuous Fréchet differentiability with respect to the shape of the scatterer of the scattered field and its corresponding far-field pattern. Our proof is based on the implicit function theorem, and assumes that the boundary of the scatterer as well as the deformation are only Lipschitz continuous. From continuous Fréchet differentiability, we deduce a stability estimate governing the variation of the far-field pattern with respect to the shape of the scatterer. We illustrate this estimate with numerical results obtained for a two-dimensional high-frequency acoustic scattering problem.

Keywords: Scattering; acoustics; Fréchet differentiability; stability; domain; derivative; Lipschitz boundary; implicit function theorem..


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