A hybrid acceleration algorithm for the computation of the static potential Green’s functions of a rectangular cavity is proposed. Similarly to Ewald’s method, it combines the series expansions in terms of images and modes. The main particularity with respect to Ewald resides in the fact that it does not need the evaluation of a non-algebraic function such as the complementary error function (erfc) while maintaining the rapid convergence of the Ewald technique. Finally, the method requires the computation of eight terms (original source plus seven images) and of several modal series corresponding to bigger cavities, which can be efficiently performed. Numerical results are provided to verify the feasibility of the algorithm, which appears as a promising alternative to the existing methods in the literature.