Journal of Applied Mathematics

Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations

Jae-Hong Pyo

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The stabilized Gauge-Uzawa method (SGUM), which is a 2nd-order projection type algorithm used to solve Navier-Stokes equations, has been newly constructed in the work of Pyo, 2013. In this paper, we apply the SGUM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We prove that SGUM is unconditionally stable, and we perform error estimations on the fully discrete finite element space via variational approach for the velocity, pressure, and temperature, the three physical unknowns. We conclude with numerical tests to check accuracy and physically relevant numerical simulations, the Bénard convection problem and the thermal driven cavity flow.

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J. Appl. Math., Volume 2013 (2013), Article ID 372906, 21 pages.

First available in Project Euclid: 14 March 2014

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Pyo, Jae-Hong. Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations. J. Appl. Math. 2013 (2013), Article ID 372906, 21 pages. doi:10.1155/2013/372906.

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  • R. H. Nochetto and J.-H. Pyo, “The gauge-Uzawa finite element method. I. The Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 43, no. 3, pp. 1043–1068, 2005.
  • R. H. Nochetto and J.-H. Pyo, “The gauge-Uzawa finite element method. II. The Boussinesq equations,” Mathematical Models & Methods in Applied Sciences, vol. 16, no. 10, pp. 1599–1626, 2006.
  • J.-H. Pyo and J. Shen, “Gauge-Uzawa methods for incompressible flows with variable density,” Journal of Computational Physics, vol. 221, no. 1, pp. 181–197, 2007.
  • J.-H. Pyo and J. Shen, “Normal mode analysis of second-order projection methods for incompressible flows,” Discrete and Continuous Dynamical Systems B, vol. 5, no. 3, pp. 817–840, 2005.
  • J. H. Pyo, “Error estimates for the second order semi-discrete stabilized Gauge-Uzawa methodFor the Navier-Stokes equations,” International Journal of Numerical Analysis & Modeling, vol. 10, pp. 24–41, 2013.
  • P. M. Gresho, R. L. Lee, S. T. Chan, and R. L. Sani, “Solution of the time-dependent incompressible Navier-Stokes and Boussinesq equations using the Galerkin finite element method,” in Approximation Methods for Navier-Stokes Problems, vol. 771 of Lecture Notes in Mathematics, pp. 203–222, Springer, Berlin, Germany, 1980.
  • K. Onishi, T. Kuroki, and N. Tosaka, “Further development of BEM in thermal fluid dynamics,” in Boundary Element Methodsin Nonlinear Fluid Dynamics, vol. 6 of Developments in Boundary Element Methods, pp. 319–345, 1990.
  • S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 1994.
  • F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, 1991.
  • V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, 1986.
  • P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, Ill, USA, 1988.
  • J. G. Heywood and R. Rannacher, “Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization,” SIAM Journal on Numerical Analysis, vol. 19, no. 2, pp. 275–311, 1982.
  • R. B. Kellogg and J. E. Osborn, “A regularity result for the Stokes problem in a convex polygon,” Journal of Functional Analysis, vol. 21, no. 4, pp. 397–431, 1976.
  • R. Temam, Navier-Stokes Equations, AMS Chelsea, 2001.