Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems
Provides a unified theoretical framework for numerical analysis of lossy Helmholtz problems, benefiting researchers in computational wave propagation.
The authors develop stability and convergence estimates for Galerkin discretizations of lossy Helmholtz problems with Robin boundary conditions, with explicit dependence on the complex wave number. The estimates unify and extend existing results for purely imaginary and real wave numbers.
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 $ζ\in\mathbb{C}$, $\operatorname{Re}ζ\geq0$, $\left\vert ζ\right\vert \geq1$. For the extreme cases $ζ\in\operatorname*{i}\mathbb{R}$ and $ζ\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.