Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

• Published in: arXiv.org 2021-02
• Main Authors: Anaya, Veronica, Caraballo, Ruben, Gomez-Vargas, Bryan, Mora, David, Ruiz-Baier, Ricardo
• Format: Article
• Language: English
• Subjects: Adaptive algorithms; Boundary conditions; Constitutive equations; Constitutive relationships; Dirichlet problem; Finite element method; Incompressibility; Mathematical analysis; Stability analysis; Subspaces; Velocity; Viscosity; Vorticity
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2102.05254

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.