Consistent Local Projection Stabilized Finite Element Methods

Abstract

This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov–Galerkin strategy for the Stokes problem. The initial unstable $\mathbb{P}_1\times\mathbb{P}_l$, $l=0,1$, finite element space is enhanced with solutions of residual-based local problems, and then the static condensation procedure is applied to derive new methods. The approach keeps degrees of freedom unchanged while giving rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure. The resulting methods do not need the use of a macroelement grid structure and are parameter-free. The numerical analysis is carried out showing optimal convergence in natural norms, and moreover, two ways of rendering the velocity field locally mass conservative are proposed. Some numerics validate the theoretical results.