The design of an Accelerator Production of Tritium (APT) facility being investigated at Los Alamos includes a linear accelerator using superconducting rf-cavities for the acceleration of a high-current cw proton beam. For electron accelerators with particles moving at the speed of light (beta approximate to 1.0), resonators with a rounded shape, consisting of ellipsoidal and cylindrical sections, are well established. They are referred to as "elliptical" cavities. For the APT-design, this shape has been adapted for much slower proton beams with beta ranging from 0.60 to 0.94. This is a new energy range, in which resonators of an elliptical type have never been used before. Simulations with the well-proven electromagnetic modeling tools MAFIA and SUPERFISH were performed. The structures have been optimized for their rf and mechanical properties as well as for beam dynamics requirements. The TRAK_RF simulation code is used to investigate potential multipacting in these structures. All the simulations will be put to a final test in experiments performed on single cell cavities that have started in our structures laboratory.