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[en] The natural element method (NEM) is extended to solve the polarized radiative transfer problem in a two-dimensional scattering medium with complex geometries, in which the angular space is discretized by the discrete-ordinates approach, and the spatial discretization is conducted by the Galerkin weighted residuals approach. The Laplace interpolation scheme is adopted to obtain the shape functions used in the Galerkin weighted residuals approach. The NEM solution to the vector radiative transfer in a square enclosure filled with a Mie scattering medium is first examined to validate our program. We then study the polarized radiative transfer in two kinds of geometries filled with scattering medium which is equivalent to a suspension of latex spheres in water. Three sizes of spheres are considered. The results for non-dimensional polarized radiative flux along the boundaries and the angular distributions of the Stokes vector at specific positions are presented and discussed. For the complex geometry bounded by the square and circular object, numerical solutions are presented for the cases both with Lambertian (diffuse) reflection and with Fresnel reflection. Some interesting phenomenon are found and analyzed. - Highlights: • Vector radiative transfer in two-dimensional scattering medium is solved by NEM. • The complex geometry bounded by the square and circular object is considered. • The circle surface is considered Lambertian reflection or Fresnel reflection. • The scattering medium is equivalent to a suspension of latex spheres in water.