  It is well known that the quasi-optimality of the Galerkin finite element
method for the Helmholtz equation is dependent on the mesh size and the
wave-number. In literature, different criteria have been proposed to ensure
quasi-optimality. Often these criteria are difficult to obtain and depend on
wave-number explicit regularity estimates. In the present work, we focus on
criteria based on T-coercivity and weak T-coercivity, which highlight mesh size
dependence on the gap between the square of the wavenumber and Laplace
eigenvalues. We also propose an adaptive scheme, coupled with a residual-based
indicator, for optimal mesh generation with minimal degrees of freedom.
