The geometrical structures of the crystalline polyethylene under several different external pressures up to 10 GPa are optimized by a pseudopotential plane wave density functional method. Both local density (LDA) and generalized gradient (GGA) approximations for exchange-correlation energy and potential are used. It is found that LDA heavily underestimate the geometry parameters under ambient pressure but GGA successfully correct them and get results in good agreements with the experimental geometry. The calculated GGA volume is about 94 Angstrom (3) in comparison with the x-ray scattering value of about 92 Angstrom (3) and the neutron scattering value of 88 Angstrom (3). The bulk and Young's modulus are calculated by means of several different methods. The Young's modulus along the chain ranges from about 350 to about 400 GPa which is in good agreement with the experimental results. But the bulk modulus is several times larger than those of experiments, indicating a different description of the interchain interactions by both LDA and GGA. The band structures are also calculated and their changes with the external pressure are discussed.