  Numerically solving the 2D Helmholtz equation is widely known to be very
difficult largely due to its highly oscillatory solution, which brings about
the pollution effect. A very fine mesh size is necessary to deal with a large
wavenumber leading to a severely ill-conditioned huge coefficient matrix. To
understand and tackle such challenges, it is crucial to analyze how the
solution of the 2D Helmholtz equation depends on (perturbed) boundary and
source data for large wavenumbers. In fact, this stability analysis is critical
in the error analysis and development of effective numerical schemes.
Therefore, in this paper, we analyze and derive several new sharp
wavenumber-explicit stability bounds for the 2D Helmholtz equation with
inhomogeneous mixed boundary conditions: Dirichlet, Neumann, and impedance. We
use Fourier techniques, the Rellich's identity, and a lifting strategy to
establish these stability bounds. Some examples are given to show the
optimality of our derived wavenumber-explicit stability bounds.
