A global residual-based stabilization for equal-order finite element approximations of incompressible flows

Due to simplicity in implementation and data structure, elements with equal‐order interpolation of velocity and pressure are very popular in finite‐element‐based flow simulations. Although such pairs are inf‐sup unstable, various stabilization techniques exist to circumvent that and yield accurate approximations. The most popular one is the pressure‐stabilized Petrov–Galerkin (PSPG) method, which consists of relaxing the incompressibility constraint with a weighted residual of the momentum equation. Yet, PSPG can perform poorly for low‐order elements in diffusion‐dominated flows, since first‐order polynomial spaces are unable to approximate the second‐order derivatives required for evaluating the viscous part of the stabilization term. Alternative techniques normally require additional projections or unconventional data structures. In this context, we present a novel technique that rewrites the second‐order viscous term as a first‐order boundary term, thereby allowing the complete computation of the residual even for lowest‐order elements. Our method has a similar structure to standard residual‐based formulations, but the stabilization term is computed globally instead of only in element interiors. This results in a scheme that does not relax incompressibility, thereby leading to improved approximations. The new method is simple to implement and accurate for a wide range of stabilization parameters, which is confirmed by various numerical examples.