# Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

## TL;DR

This paper develops a stability and convergence theory for Galerkin discretizations of the lossy Helmholtz equation, providing explicit estimates that depend on the complex wave number and smoothly transition between different regimes.

## Contribution

It introduces a wave number-explicit analysis framework for the lossy Helmholtz problem with Robin boundary conditions, unifying and extending existing estimates.

## Key findings

- Explicit stability and convergence estimates for all complex wave numbers considered.
- Seamless transition of estimates between purely imaginary and real wave numbers.
- Unified analysis that covers boundary cases and general complex wave numbers.

## 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 part of the complex wave number $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert \zeta\right\vert \geq1$. For the extreme cases $\zeta \in\operatorname*{i}\mathbb{R}$ 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.