Restore optimal cubic L2 convergence for pressure-robust Taylor–Hood elements

Determine a remedy that restores the cubic L2-velocity convergence rate for the pressure-robust Taylor–Hood finite element family, whose commuting-diagram-constrained Nédélec interpolation currently causes a loss of one convergence order relative to the standard non-pressure-robust method.

Background

The paper applies its force-interpolation technique to the Taylor–Hood pair with velocity space [L{k+1}(\mathcal{T}_h)]2 and pressure space Lk(\mathcal{T}_h). Pressure-robustness requires using the Nédélec interpolation operator N{k-1}, because this is the operator that commutes with the Lagrange interpolation operator associated with the pressure space.

For the lowest-order Taylor–Hood pair P2P1, the resulting pressure-robust method achieves optimal quadratic convergence in the H1 velocity norm and pressure norm. However, its L2 velocity estimate is only quadratic, whereas the standard non-pressure-robust P2P1 method achieves cubic convergence. The order loss arises because the interpolation operator is tied to the lower-degree pressure space, and the authors state that no choice of k within the Taylor–Hood family improves this limitation.

References

In the $L2$ norm, however, we only get the bound

|u-u_h|{L2} \lesssim h2\,(h|u|{H3}+|\Delta u|_{H2}),

which is only quadratic, one order below the cubic rate of the standard, non-pressure-robust $P2P1$ method: the term $\nu(\Delta u - N{0}(\Delta u), Q{1} w)$ - following the same steps as in the proof of \Cref{thm:mini_l2} - is now only of order $h2$ by \Cref{prop:consistency_low}. Since $N{k-1}$ is tied to the pressure space by the commuting diagram, no choice of $k$ can improve on this loss of one order for the Taylor--Hood family. So far we have not been able to find a remedy for this, and we leave it as an open problem.

— Pressure-robustness by commuting interpolation operators for Stokes discretizations with continuous pressures  (2609.18657 - Lederer et al., 16 Sep 2026) in Section 5, “The Taylor--Hood element: a cautionary example” (following the displayed L2-error estimate)