An adaptive conforming mixed method with high-order elements for clustered eigenvalues of the Helmholtz transmission problem with index of refraction n(x)=1

In this paper, we investigate the conforming mixed method for the Helmholtz transmission eigenvalue problem in cases where the scatterers have the same permeability in 2D or the same sound speed in 3D as the surrounding medium. Based on the work of Cakoni et al. (2009) [22] and Liu et al. (2023) [23], we study the approximation of high-order elements for the clustered eigenvalues of the problem. We present an a priori error estimate, derive an a posteriori error estimator, and prove the reliability of the estimator for the approximate eigenvalues and the approximate eigenspaces. Numerical experiments show that our method is efficient and can compute the approximate eigenvalues with high precision.