Method of lines and finite element analysis investigations have been performed to optimize parameters in a TE/sub 011/ mode cavity resonator suitable for a spaceborne hydrogen maser. We report on designs that were explored to find a global maximum in the important design parameters for the microwave cavity used in a hydrogen maser. The criteria sought in this exercise were both the minimization of the total volume of the cavity and the maximization of the product of the z-component of the magnetic energy filling factor and the cavity TE/sub 011/ mode Q-factor (Q/spl middot//spl eta/). Different configurations were studied. They were a sapphire tube in a copper cylinder, a sapphire tube in a copper cylinder with Bragg reflectors, and spherical copper cavities both empty and sapphire-lined on the inside cavity surface. At 320 K, the simulations resulted in an optimum product Q/spl middot//spl eta/ = 4.9 /spl times/ 10/sup 4/, with an inner cavity radius of 80 mm and unity aspect ratio. This represents a 54% improvement over an earlier design. The expected increase in the product Q/spl middot//spl eta/ with the inclusion of Bragg reflectors to the sapphire tube was not achieved. Moreover, the z-component of the magnetic energy filling factor was greatly reduced due to an increase in the radial magnetic field. The sapphire-lined spherical cavity showed no better performance than an equivalent-sized empty copper spherical cavity. For the empty cavity the simulations resulted in the product Q/spl middot//spl eta/ = 4.4 /spl times/ 10/sup 4/. The empty spherical cavity resonator is not suitable for the spaceborne hydrogen maser as the total volume in this case is 33% larger than that of the optimized sapphire tube resonator.