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This work is devoted to the finite element discretization of the incompressible Navier--Stokes equations. The starting point is a low order stabilized finite element method using piecewise linear continuous discrete velocities and piecewise constant pressures. This pair of spaces needs to be stabilized, and, as such, the continuity equation is modified by adding a stabilizing bilinear form based on the jumps of the pressure. This modified continuity equation can be rewritten in a standard way involving a modified different velocity field, which is as a consequence divergence-free. This modified velocity field is then fed back to the momentum equation making the convective term skew-symmetric. Thus, the discrete problem can be proven stable without the need to rewrite the convective field in its skew-symmetric way. Error estimates with constants independent of the viscosity are proven. Numerous numerical experiments confirm the theoretical results.

In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier--Stokes equations with singular sources on Lipschitz, but not ne...

We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control proble...

This work is devoted to the proposal, analysis, and numerical testing of a fully computable a posteriori error bound for a class of nonlinear discretizations of the convection-diffusion-reaction equation. The type of discretization we consider is nonlinear, since it has been built with the aim of preserving the discrete maximum principle. Under mild assumptions on the stabilizing term, we obtain an a posteriori error estimator that provides a certified upper bound on the norm of the error. Under the additional assumption that the stabilizing term is both Lipschitz continuous and linearity preserving, the estimator is shown to be locally efficient. We present examples of discretizations that satisfy these two requirements, and the theory is illustrated by several numerical experiments in two and three space dimensions.