We present the firm application of the semi-implicit semi-Lagrangian integration technique to a finite-element barotropic model using the shallow-water equations on a variable-resolution. Two schemes based on this approach, differing only in their treatment of the rotational part of the wind field, are formulated and analyzed. A set of comparative experiments was performed using carefully balanced initial conditions (to eliminate spurious high-frequency oscillations); an Eulerian control integration was run at high resolution on a uniform grid. Both schemes are stable with timesteps at least six times longer than the limiting timestep of the corresponding Eulerian scheme using the same variable-resolution mesh. However, one scheme is consistently more accurate than the other. These results were explained by a theoretical analysis of the stability and accuracy of the schemes. We conclude that the semi-implicit semi-Lagrangian scheme is a promising technique for finite-element models as well as for finite-difference models.