Multigrid V-cycle solution of the Poisson equation for self-gravity in DIFFHYDRO. At the finest level ℓ = 1, the gravitational potential ϕ^h is computed by solving A^hx^h = b^h with residual r^h = b^h − A^hx^h. The residual is restricted to successively coarser meshes (ℓ = 2, 3), where the error equations A^2he^2h = r^2h and A^4he^4h = r^4h are solved, using a direct solver on the coarsest grid. The coarse-grid error is then interpolated/prolongated back to finer levels and used to update the solution, x^h ← x^h + e^h, iterating until convergence. The bottom panels illustrate the density field represented on each level, highlighting how coarse grids capture long-wavelength contributions efficiently and accelerate convergence of the self-gravity force used in the hydrodynamic evolution.