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

TL;DR

This paper introduces a pressure-robust, optimally convergent H(div) finite element method for the steady Oseen equations that does not rely on the commuting diagram property, enhancing efficiency and robustness in convection-dominated flows.

Contribution

It develops a convergence theory for H(div)-conforming methods without the commuting diagram property, incorporating vorticity stabilization and demonstrating pressure robustness and computational advantages.

Findings

Achieves optimal velocity error convergence independent of the discrete inf-sup constant.

Demonstrates pressure robustness and fewer degrees of freedom with Stenberg finite elements.

Applicable to finite element pairs violating the commuting diagram property, suitable for high Reynolds number flows.

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.