The double-spherical harmonics method (DPN) is a common approximation in the study of the neutron transport equation in reactor physics problems. Inside a reactor near points where strong discontinuities in material properties occur, such as bare boundaries or areas near strong absorbers, there is usually greater anisotropy in the angular distribution of neutron flux. A more appropriate description of the angular flux behavior at such points requires the use of methods such as DPN, which utilize separate expansions for different directions of neutron motion instead of using a single expansion for all directions, as in the PN method. In this paper, the multigroup, one-dimensional neutron transport equation in the Cartesian coordinate system is solved using the DP1 approximation. To do this, first the multigroup DP1 equations and the corresponding boundary conditions are derived, then they are written in the form of multigroup neutron diffusion equations, which are here called the simplified-DP1 or SDP1 equations. The finite element method is then used to numerically solve the SDP1 equations. The results of the proposed method are discussed for several different test problems in comparison with the P3 method.