Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure-robust, in the sense that the velocity H1-error is proportional to the best velocity H1-error. This shows that such a property can be achieved without using conforming and divergence-free pairs. We also bound the pressure L2-error, only in terms of the best velocity H1-error and the best pressure L2-error. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by discontinuous Galerkin methods.