The Sharp Phase Field Method (S-PFM) is a phase field approach in which diffuse interfaces that delimit precipitates or phases in evolving microstructures are resolved with only one grid point without breaking continuous translational invariance, allowing to multiply accessible linear dimensions by an order of magnitude or, conversely, to reduce computational times by almost three order of magnitudes. However, in most microstructures of interest, elastic fields are involved (precipitates with specific eigenstresses, cracks, voids, dislocations...). The modeling of these microstructures with sharp interfaces requires the use of accurate mechanical solvers, able to handle infinite material contrasts, with no overshoot and no oscillations near the interfaces. We therefore developed new discrete mechanical solvers based on FFT that are mathematically stable and thus free of any instability, even in the presence of infinite elastic contrasts or stress singularities, and that are computationally very efficient. We will illustrate the ability of these solvers to accurately reproduce mechanical fields in a variety of situations, including voids, cracks, precipitates and composite materials.