# Wavenumber explicit analysis for Galerkin discretizations of lossy Helmholtz problems

Melenk, Jens Markus; Sauter, Stefan A; Torres, Céline (2020). Wavenumber explicit analysis for Galerkin discretizations of lossy Helmholtz problems. SIAM Journal on Numerical Analysis, 58(4):2119-2143.

## 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.

## 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.

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