The high-frequency pressure induced by a surface acoustic wave in the fluid filling a microchannel is computed by solving the full scattering problem. The microchannel is fabricated inside a container attached to the top of a piezoelectric substrate where the surface wave propagates. The finite element method is used. The pressure found in this way is compared with the pressure obtained by solving boundary-value problems formulated on the basis of simplifications which have been introduced in earlier papers by other research studies. The considered example shows that the difference between the results can be significant, ranging from several tens of percent up to several times in different points inside the channel.