Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

We show that the discrete approximate volume preserving mean curvature flow in flat torus $$\mathbb {T}^N$$ starting near a strictly stable critical set E of perimeter converges long time to translate exponentially fast. As an intermediate result we establish new quantitative estimate Alexandrov type for periodic constant hypersurfaces. Finally, two dimensional case complete characterization behaviour with arbitrary initial sets finite is provided.