A finite-element procedure is presented employing both self-adjoint and non-self-adjoint variational principles associated with the neutron-transport equation. This scheme admits independent approximation of the even-and odd-parity components of the angular flux in any subregion of the system under consideration. Numerical examples in plane and spherical geometries demonstrate that in some cases a substantial gain in accuracy is achieved by the new scheme as compared to the self-adjoint approach.