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.

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有损亥姆霍兹问题伽辽金离散化的波数显式分析

我们提出了有损亥姆霍兹方程及其伽辽金离散化的稳定性和收敛理论。边界条件是罗宾型的。所有估计对于复波数的实部和虚部都是明确的 $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert \zeta\right \vert \geq1$。对于极端情况 $\zeta\in\operatorname*{i}\mathbb{R}$ 和 $\zeta\in\mathbb{R}_{\geq0}$，估计值与文献中的现有估计值一致，并且在右复半平面中展示这些情况之间的无缝过渡。