On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier–Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix–Raviart element is proposed, where divergence-free, lowest-order Raviart–Thomas velocity reconstructions reestablish L2L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.