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.

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