On the Reynolds-number scaling of Poisson solver complexity

We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations.