We aim to answer the following question: is the complexity of numerically solving Poisson's 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 Re-number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations.
Trias et al. (2026) studied this question.