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，针对亥姆霍兹方程的伽辽金有限元方法的准最优性依赖于网格尺寸和波数。文献中提出了多种确保准最优性的判据，但这些判据通常难以获取，且依赖于波数显式正则性估计。本文聚焦于基于T- 共轭和弱T- 共轭的判据，揭示了网格尺寸与波数平方及拉普拉斯特征值间隙之间的依赖关系。我们同时提出一种自适应方案，结合残差型指示器，以最少自由度实现最优网格生成。