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Convolution quadrature for the wave equation with a nonlinear impedance boundary condition

Authors: Lehel Banjai and Alexander Rieder
Journal: Math. Comp. 87 (2018), 1783-1819
MSC (2010): Primary 65M38, 65M12, 65R20
Published electronically: October 4, 2017
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Abstract: A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory.

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Additional Information

Lehel Banjai
Affiliation: The Maxwull Institute for Mathematical Sciences, School of Mathematical & Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

Alexander Rieder
Affiliation: Institute for Analysis and Scientific Computing (Inst. E 101), Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Keywords: Convolution quadrature method, boundary element method, time domain, wave propagation
Received by editor(s): April 13, 2016
Received by editor(s) in revised form: February 1, 2017
Published electronically: October 4, 2017
Additional Notes: This research was supported by the Austrian Science Fund (FWF) through the doctoral school “Dissipation and Dispersion in Nonlinear PDEs” (project W1245, A.R.).
Article copyright: © Copyright 2017 American Mathematical Society

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