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.

翻译：众所周知，Helmholtz方程的Galerkin有限元法的拟最优性依赖于网格尺寸和波数。已有文献提出了不同的准则来确保拟最优性，但这些准则通常难以获得且依赖于波数显式的正则性估计。本文聚焦于基于T- coercivity和弱T- coercivity的准则，这些准则凸显了网格尺寸与波数平方及Laplace特征值间隙的依赖关系。我们进一步提出了一种结合残差型指示器的自适应方案，用于以最小自由度生成最优网格。