An efficient Ritz method is developed from the variational principle to solve the Poisson-Boltzmann equation under the Debye-Huckel approximation for studying the EOF in microchannels. The method is applied to the family of superelliptic cross sections which includes the elliptic channel and the rectangular channel as limiting cases. Several accurate tables presented are useful for design of electroosmotic channels, especially rectangular channels with rounded corners. It is shown how the flow rate Q is a sophisticated consequence of the nondimensional electrokinetic width K, the aspect ratio b as well as the superelliptic exponent n.