  The transmission eigenvalue problem arising from the inverse scattering
theory is of great importance in the theory of qualitative methods and in the
practical applications. In this paper, we study the transmission eigenvalue
problem for anisotropic inhomogeneous media in $\Omega\subset
\mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation
theory, we derive an a posteriori estimator of residual type and prove its
effectiveness and reliability for eigenfunctions. In addition, we also prove
the reliability of the estimator for transmission eigenvalues. The numerical
experiments indicate our method is efficient and can reach the optimal order
$DoF^{-2m/d}$ by using piecewise polynomials of degree $m$ for real
eigenvalues.
