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This paper presents a formulation and some calculations for the three dimensional analysis of RF electromagnetic fields by using the finite element method. We adopted the penalty method to obtain an approximate solution of the stationary Maxwell equations. The discussion is focused on the analysis of electromagnetic waves as a linear eigenvalue problem, for which a primitive variable approach based on the electric field, E, is given. We confirmed the validity of this method by two dimensional calculations and developed a three dimensional calculation code MAX3D. Satisfactory results are obtained for rectangular cavities, cylindrical cavities and those cavities with smoothly deformed parts.