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

• Verónica Anaya , David Mora , Amiya K. Pani and Ricardo Ruiz-Baier

## 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.

MSC 2010: 65N30; 65N12; 76D07; 65N15

## Acknowledgment

The authors are grateful to Prof. Maxim Olshanskii for the stimulating discussions regarding Bernoulli pressure formulations. This work has been partially supported by DICREA-UBB through projects 2020127 IF/R and 2120173 GI/C, by ANID-Chile through Centro de Modelamiento Matemático (FB210005), Anillo of Computational Mathematics for Desalination Processes (ACT 210087), Fondecyt project 1211265, by the HPC-Europa3 Transnational Access programme, and by the Ministry of Science and Higher Education of the Russian Federation within the framework of state support for the creation and development of World-Class Research Centers ‘Digital biodesign and personalised healthcare’ No. 075-15-2022-3046.

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