The natural element method is used to simulate two-dimensional viscous flows where interfacial effects must be taken into account, for application to polymer melts at the micro and nano scales. The variational formulation includes surface tension on the free surfaces, a net wetting force is applied at the contact line where the fluid reaches a solid surface, and the Navier-slip condition is used along fluid-solid interfaces. No dynamic wetting angle is prescribed, and the contact angle obtained results from the other material parameters and from overall flow conditions. A comparison with an analytical solution in a simple surface tension-driven flow is given, and contact with a rigid solid is involved in the transient spreading of a droplet and in the steady movement of a meniscus between two plates.