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Standard mixed finite element methods for the incompressible Navier–Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to infsup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H and L errors of the velocity and the L error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results. Introduction In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations are balanced completely by the pressure gradient. Unfortunately, nearly all mixed discretisations [3,15] for the incompressible Navier–Stokes equations in the primal variables velocity and pressure do not preserve this property exactly which excites in some flow problems the so-called numerical instability of poor mass conservation [10, 13, 14, 18, 23]. However, poor mass conservation is just the unwanted companion of an idea which is usually regarded as a success story: the relaxation of the divergence constraint in mixed methods for incompressible flows. While this numerical instability is traditionally mitigated by stabilisation techniques like the grad-div stabilisation [4, 12, 17, 25, 26, 28], a novel strategy was recently proposed in [19]. The key observation is that stabilisation issues in inf-sup stable mixed methods arise only due to a relaxation of the divergence constraint in the discrete velocity test functions, and not in the trial functions. Moreover, stabilisation

Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors