A finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient

TL;DR

This paper introduces a novel finite element method for incompressible viscous flow based on minimizing the pressure gradient, eliminating pressure degrees of freedom, and enabling accurate, stable solutions on coarse meshes.

Contribution

It develops a pressure-minimizing variational formulation discretized with Q9 elements, enforcing incompressibility via constraints and providing wall forces without pressure reconstruction.

Findings

Accurately recovers Poiseuille flow at machine precision

Achieves convergence rate of approximately 3.3 on Kovasznay flow

Produces smooth, oscillation-free solutions on coarse meshes in convection-dominated regimes

Abstract

We present a finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient (PMPG). This variational principle, recently established by Taha, Gonzalez & Shorbagy (Phys. Fluids, vol. 35, 2023), states that the Navier-Stokes equations are equivalent to determining the rate of change of velocity at each instant by minimizing the L2 norm of the implied pressure gradient, subject to incompressibility and boundary conditions. We discretize the PMPG functional directly using Q9 biquadratic finite elements and minimize over the nodal velocity rates (Rayleigh-Ritz). No pressure degrees of freedom appear; incompressibility and boundary conditions are enforced as linear equality constraints through a monolithic saddle-point system, whose Lagrange multipliers provide wall forces without pressure reconstruction. We verify the formulation against exact Poiseuille flow (machine-precision recovery) and the Kovasznay solution (convergence rate ~3.3), and validate it against published benchmarks for the lid-driven cavity, the backward-facing step, and flow past a circular cylinder. The formulation produces smooth, oscillation-free solutions on coarse meshes in the convection-dominated regime without stabilization. We further show that the element-wise PMPG functional density serves as a built-in error indicator for adaptive mesh refinement, and that the stationarity condition can be read backwards to estimate the kinematic viscosity directly from velocity field measurements.