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This paper applies the Recursive Projection Method (RPM) to the problem of finding the effective mechanical response of a periodic heterogeneous solid. Previous works apply the Fast Fourier Transform (FFT) in combination with various fixed-point methods to solve the problem on the periodic unit cell. These have proven extremely powerful in a range of problems ranging from image-based modeling to dislocation plasticity. However, the fixed-point iterations can converge very slowly, or not at all, if the elastic properties have high contrast, such as in the case of voids. The paper examines the reasons for slow, or lack of convergence, in terms of a variational perspective. In particular, when the material contains regions with zero or very small stiffness, there is lack of uniqueness, and the energy landscape has flat or shallow directions. Therefore, in this work, the fixed-point iteration is replaced by the RPM iteration. The RPM uses the fixed-point iteration to adaptively identify the subspace on which fixed-point iterations are unstable, and performs Newton iterations only on the unstable subspace, while fixed-point iterations are performed on the complementary stable subspace. This combination of efficient fixed-point iterations where possible, and expensive but well-convergent Newton iterations where required, is shown to lead to robust and efficient convergence of the method. In particular, RPM-FFT converges well for a wide range of choices of the reference medium, while usual fixed-point iterations are usually sensitive to this choice.