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Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, wesolve the Maxwell's transmission eigenvalue problem. Both real and complexeigenvalues are considered. Based on the fixed-point weak formulation withreasonable assumptions, the optimal error estimates for numerical eigenvaluesand eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bfH}(\mathrm{curl})$-semi-norm) are established.

$\bf H(\mathrm{curl}^2)$ conforming element for Maxwell's transmission eigenvalue problem using fixed-point approach