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A fully high-order finite element algorithm is presented for simulating electromagnetic scattering by a deep cavity using curvilinear tetrahedral elements. Unlike the previously developed algorithm based on mixed-order triangular prism elements, this new algorithm can significantly suppress numerical dispersion errors in all directions, thus resulting in a more accurate and efficient simulation of complex wave propagation inside a deep cavity. Moreover, the use of tetrahedral elements removes limitations on the modeling of complex cavities imposed by the use of triangular prism elements. Numerical examples and experimental results are presented to demonstrate the advantages of the new algorithm.