Error analysis for a vorticity/Bernoulli pressure formulation for the Oseen equations

Abstract

Abstract A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L2-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an a posteriori error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and also confirm the theoretical findings.