A Communication-Free Parallel Screened Poisson Solver for Incompressible Navier–Stokes

The scalability of incompressible Navier–Stokes solvers on massive parallel clusters is fundamentally constrained by the pressure projection step, where the global elliptic coupling of the standard Pressure Poisson Equation (PPE) necessitates expensive all-to-all communication and creates a severe latency bottleneck. To overcome this barrier, we introduce Locality-Certified Screened Projection (LCSP), a novel framework enabling a fully parallel, communication-free pressure solve. By relaxing the strict incompressibility constraint into a penalized form ∇· uⁿ⁺¹ + ηψ = 0, we transform the PPE into a screened Helmholtz problem (-Δ + κ²) ψ = ƒ. This operator exhibits intrinsic locality characterized by the exponential Yukawa decay of its Green's function. Leveraging this property, we implement a single-pass Overlap-Restrict assembly strategy: the computational domain is partitioned into overlapping tiles where local problems are solved entirely independently, and solutions are then restricted to the core without any inter-subdomain trace exchange. Our rigorous error analysis demonstrates that artifacts from artificial tile boundaries decay exponentially with the overlap width, allowing the mass conservation error to be explicitly controlled via the screening parameter κ. Extensive numerical benchmarks confirm that LCSP successfully decouples the global dependency, reduces the peak memory footprint to Ο(|tile|), and achieves optimal linear weak scaling for large-scale flow simulations. Ultimately, LCSP establishes a mathematically grounded trade-off between exact incompressibility and parallel efficiency, providing a robust, highly scalable solution for high-fidelity CFD simulations on next-generation heterogeneous supercomputers.