Optimal L² Velocity Error Estimates for a Modified Pressure-Robust Crouzeix-Raviart Stokes Element

Abstract

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix–Raviart element such that its velocity error becomes pressure independent. This modified robust Crouzeix–Raviart element employs lowest-order Raviart–Thomas elements in a variational crime in the right-hand side of the Stokes discretization, reestablishing the $L^2$-orthogonality of divergence-free and irrotational forces in the momentum balance. The modification results in an $\mathcal{O}(h)$ consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the $L^2$ norm of the pressure. However, though the optimal convergence of the velocity in the $L^2$ norm was observed numerically, it has not yet been proven for the lowest-order Raviart–Thomas elements as the proof available for higher-order elements is not applicable in this case. In this contribution, this gap is closed and the observed $L^2$ convergence rate for the velocities is shown for the lowest-order Raviart–Thomas elements. Moreover, the dependence of the energy error estimates on the discrete inf–sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in two and three dimensions illustrate the theoretical findings