Numerical analysis of a new formulation for the Oseen equations in terms of vorticity and Bernoulli pressure

TL;DR

This paper introduces a new variational formulation for the Oseen equations using vorticity and Bernoulli pressure, along with a finite element method, error analysis, and adaptive refinement, validated through numerical experiments.

Contribution

The paper presents a novel formulation and finite element approach for the Oseen equations in terms of vorticity and Bernoulli pressure, including error analysis and adaptive methods.

Findings

Error estimates in the L^2 norm for vorticity, pressure, and velocity.

Robust and efficient a posteriori error estimator.

Numerical experiments confirming theoretical results and adaptive refinement effectiveness.

Abstract

A variational formulation is introduced for the Oseen equations written in terms of vor\-ti\-city 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 N\'ed\'elec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The {\it a priori} error analysis is carried out in the $\mathrm{L}^2$-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an {\it 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 they also confirm the theoretical findings.