Mesh-to-mesh data transfer methods are evaluated in terms of not only accuracy but also the cost of computing time. Interpolation is a widely used method because of its simplicity. However, it suffers from a lack of precision when Lagrange elements are considered. Projection methods, which are more complex, are thus preferred, although they significantly increase the computational time as well as the memory cost. We propose a comparison between these two methods (interpolation and projection) applied to curl-conforming and div-conforming elements. It appears that interpolation is advantageous because of its simplicity and speed while giving results of the same order as the projection method. A process to accurately compute circulations along edges and fluxes through faces is proposed. It relies on an adaptive quadrature scheme which provides reliable results regardless of the variation of the initial solution.