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

Abstract

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 Babuška-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.

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Acknowledgements

This work has been partially supported by DIUBB through projects 2020127 IF/R and 194608 GI/C; by the National Agency for Research and Development, ANID-Chile, through projects FONDECYT 1211265 and “Centro de Modelamiento Matemático” (AFB170001) of the PIA Program: “Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal”; by the Monash Mathematics Research Fund S05802-3951284; by the HPC-Europa3 Transnational Access programme through grant HPC175QA9K; 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 personalized healthcare” No. 075-15-2020-926.

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Correspondence to Verónica Anaya.

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Anaya, V., Caraballo, R., Gómez-Vargas, B. et al. Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity. Calcolo 58, 44 (2021). https://doi.org/10.1007/s10092-021-00433-6

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  • DOI: https://doi.org/10.1007/s10092-021-00433-6

Keywords

  • Oseen equations
  • Velocity-vorticity-pressure formulation
  • Mixed finite element methods
  • Variable viscosity
  • A priori and a posteriori error analysis
  • Adaptive mesh refinement

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 76D07
  • 65N15