The authors describe a Galerkin method (GM), which determines efficiently the basis functions satisfying the edge conditions and the size of the elements on the Neumann boundaries in this paper. The calculated higher order mode cutoff frequencies in both symmetric and asymmetric TEM cells by the GM are presented. The authors could then solve the technical debate for TE12-, TE03-, TM11-, and TM12-mode cutoff frequencies in the symmetric TEM cell. It is also shown that the measured resonant frequencies of a designed asymmetric TEM cell agree well with the predicted data by the GM.