Transmission eigenvalues play an important role in the inverse scattering theory. In this paper, we study the mixed discontinuous Galerkin method for the transmission eigenvalue problem and the modified transmission eigenvalue problem for anisotropic inhomogeneous medium in Ω⊂Rd(d=2,3)\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset {\mathbb {R}}^d\,(d=2,3)$$\end{document}. We use the T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}$$\end{document}-coercivity, Gårding’s inequality, the consistency of the DG method and the compact embeddings of broken Sobolev spaces to prove that the discrete solution operator Kh\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {K}}_{h}$$\end{document} converges pointwise to the solution operator K\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {K}}$$\end{document} and {Kh}\documentclass[12pt]{minimal}
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\begin{document}$$\{{\mathbb {K}}_{h}\}$$\end{document} is collectively compact. Then we employ the spectral approximation theory and the approximation property of the DG finite element space to prove the hp a priori error estimate of approximate eigenpairs.
