Pressure-robust optimally convergent H(div) finite element method without the commuting diagram property for the steady Oseen equations

Abstract: This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The proposed method incorporates vorticity stabilization to ensure optimal-order convergence of the velocity error, especially for convection-dominated cases. As a crucial component of the analysis, exact de Rham and finite element complexes provide a framework whose utility includes establishing velocity error estimates independent of the discrete inf-sup constant. As a representative example, Stenberg finite elements demonstrate the framework's validity and offer several computational advantages: pressure robustness, fewer degrees of freedom than classical RT or BDM elements due to vertex continuity, and convergence without requiring the commuting diagram property. Moreover, the proposed methodology is applicable to a class of finite element pairs that violate the commuting diagram property, thereby offering new possibilities for efficient discretizations of incompressible fluid problems, particularly in high Reynolds number regimes.