A numerical study on the stability of a class of Helmholtz problems

This paper concerns the stability of a class of Helmholtz problems in rectangular domains. A well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane. Error analysis of numerical methods for cavity problems relies heavily on the stability estimates. However, it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers. In this paper a high-order finite element approximation is proposed for calculating the stability bound. Numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense. Our numerical results also help to understand the stability of the multi-frequency inverse problems.