An adaptive mesh refinement strategy to ensure quasi-optimality of finite element methods for self-adjoint Helmholtz problems

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 the literature, different criteria have been proposed to ensure uniform quasi-optimality of the discretisation. In the present work, we study the uniform quasi-optimality of $H^1$ conforming and non-conforming Crouzeix-Raviart discretisation of the self-adjoint Helmholtz problem. In particular, we propose an adaptive scheme, coupled with a residual-based indicator, for generating guaranteed quasi-optimal meshes with minimal degrees of freedom.