Abstract

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary parts of the complex wavenumber $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert \zeta\right\vert \geq1$. For the extreme cases $\zeta \in{\rm i} \mathbb{R}$ and $\zeta\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.

Keywords

  1. Helmholtz equation
  2. stability
  3. $hp$-finite elements

MSC codes

  1. 35J05
  2. 65N30
  3. 65N12

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2119 - 2143
ISSN (online): 1095-7170

History

Submitted: 2 April 2019
Accepted: 20 April 2020
Published online: 20 July 2020

Keywords

  1. Helmholtz equation
  2. stability
  3. $hp$-finite elements

MSC codes

  1. 35J05
  2. 65N30
  3. 65N12

Authors

Affiliations

Funding Information

Swiss National Science Foundation : 172803
Austrian Science Fund https://doi.org/10.13039/501100002428 : W1245

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