A New Variational Multiscale FEM for the Steady-State Natural Convection Problem with Bubble Stabilization

REFERENCES

• I. Christie and A. R. Mitchell Upwinding of High Order Galerkin Methods in Conduction-convection Problems, Int. J. Numer. Meth. Eng., vol. 12, pp. 1764–1771, 1978.
   Web of Science ®Google Scholar
• J. Boland and W. Layton An Analysis of the Finite Element Method for Natural Convection Problems, Numer. Meth. Part. Differ. Equation, vol. 2, p. 115–126, 1990.
   Google Scholar
• J. Boland and W. Layton Error Analysis for Finite Element Methods for steady natural convection problems, Numer. Funct. Anal. Optim., vol. 11, p. 449–483, 1990.
   Web of Science ®Google Scholar
• A. Russo Bubble Stabilization of Finite Element Methods for the Linearized Incompressible Navier-Stokes Equations, Comput. Meth. Appl. Mech. Eng., vol. 132, p. 335–343, 1996.
   Web of Science ®Google Scholar
• A. Çıbık and S. Kaya A Projection-based Stabilized Finite Element Method for Steady-state Natural Convection Problem, J. Math. Anal. Appl., vol. 381, p. 469–484, 2011.
   Web of Science ®Google Scholar
• Z. Luo, J. Zhu, Z. Xie, and G. Zhang A Difference Scheme and Numerical Simulation Based on Mixed Finite Element Method for Natural Convection Problem. Appl. Math. Mech. Eng. Ed., vol. 24, p. 973–983, 2003.
   Google Scholar
• Z. Luo Theory Bases and Applications of Finite Element Methods. Science Press, Beijing, 2006 (in Chinese).
   Google Scholar
• P. Huang, T. Zhang, and Z. Si A Stabilized Oseen Iterative Finite Element Method for Stationary Conduction-Convection Equations, Math. Meth. Appl. Sci., vol. 35, p. 103–118, 2012.
   Web of Science ®Google Scholar
• P. Huang, X. Ma An Oseen Iterative Finite-element Method for Stationary Conduction-Convection Equations, Int. J. Comput. Math., vol. 89, p. 217–230, 2012.
   Web of Science ®Google Scholar
• P. Huang, X. Feng, and Y. He A Quadratic Equal-Order Stabilized Finite Element Method for the Conduction-Convection Equations, Comput. Fluids, vol. 86, pp. 169–176, 2013.
   Web of Science ®Google Scholar
• L. Qian, H. Cai, R. Guo, and X. Feng The Characteristic Variational Multiscale Method for Convection-dominated Convection-diffusion-reaction Problems. Int. J. Heat Mass Transfer, vol. 72, p. 461–469, 2014.
   Web of Science ®Google Scholar
• H. Su, D. Gui, P. Huang, and X. Feng Two-Level Stabilized, Nonconforming Finite-Element Algorithms for the Stationary Conduction-Convection Equations, Numer. Heat Transfer, B: Fund., vol. 66(3), p. 211–242, 2014.
   Web of Science ®Google Scholar
• H. Su, L. Qian, D. Gui, and X. Feng Second Order Fully Discrete and Divergence Free Conserving Scheme for Time-Dependent Conduction-Convection Equations, Int. Commun. Heat Mass., vol. 59, p. 120–129, 2014.
   Web of Science ®Google Scholar
• H. Su, J. Zhao, D. Gui, and X. Feng Two-Level Defect-Correction Oseen Iterative Stabilized Finite Element Method for the Stationary Conduction-Convection Equations, Int. Commun. Heat Mass., vol. 56, p. 133–145, 2014.
   Web of Science ®Google Scholar
• B. Du, H. Su, and X. Feng Two-level Variational Multiscale Method Based on the Decoupling Approach for the Natural Convection Problem, Int. Commun. Heat Mass., vol. 61, pp. 128–139, 2015.
   Web of Science ®Google Scholar
• T. J. R. Hughes, G. R. Feijo, L. Mazzei, and J. Quincy The Variational Multiscale Method a Paradigm for Computational Mechanics, Comput. Meth. Appl. Mech. Eng., vol. 166, p. 3–24, 1998.
   Web of Science ®Google Scholar
• T. J. R. Hughes, L. Mazzei, and K. E. Jansen Large Eddy Simulation and the Variational Multiscale Method, Comput. Vis. Sci., vol. 3, p. 47–59, 2000.
   Google Scholar
• T. J. R. Hughes, L. Mazzei, and A. A. Oberai The Multiscale Formulation of Large Eddy Simulation: Decay of Homogeneous Isotropic Turbulence, Phys. Fluids, vol. 13, p. 505–511, 2001.
   Web of Science ®Google Scholar
• V. John and S. Kaya A Finite Element Variational Multiscale Method for the Navier-Stokes Equations, SIAM J. Sci. Comput., vol. 26, p. 1485–1503, 2005.
   Web of Science ®Google Scholar
• V. John, S. Kaya, and W. Layton A Two-Level Variational Multiscale Method for Convection-Dominated Convection-Diffusion Equations, Comput. Meth. Appl. Mech. Eng., vol. 195, p. 4594–4603, 2006.
   Web of Science ®Google Scholar
• S. Kaya and B. Rivière A Two-Grid Stabilization Method for Solving the Steady-State Navier-Stokes Equations, Numer. Meth. Part. Differ. Equation, vol. 22, p. 728–743, 2005.
   Web of Science ®Google Scholar
• A. Masud and R. A. Khurram A Multiscale Finite Element Method for the Incompressible Navier-Stokes Equations, Comput. Meth. Appl. Mech. Eng., vol. 195, p. 1750–1777, 2006.
   Web of Science ®Google Scholar
• H. B. Zheng, Y. R. Hou, F. Shi, and L. N. Song A Finite Element Variational Multiscale Method for Incompressible Flows Based on Two Local Gauss Integrations, J. Comput. Phys., vol. 228, p. 5961–5977, 2009.
   Web of Science ®Google Scholar
• R. Araya, G. R. Barrenechea, and F. Valentin Stabilized Finite Element Method Based on Multiscale Enrichment for the Stokes Problem, SIAM J. Numer., vol. 44, p. 322–348, 2006.
   Web of Science ®Google Scholar
• Z. Ge and J. Yan Analysis of Multiscale Stabilized Finite Element Method for the Stationary Navier-Stokes Equations, Nonlinear Anal. RWA., vol. 13, p. 385–394, 2012.
   Web of Science ®Google Scholar
• J. Wen and Y. He Convergence Analysis of a New Multiscale Finite Element Method for the Stationary Navier-Stokes Problem, Comput. Math. Appl., vol. 67, p. 1–25, 2014.
   Web of Science ®Google Scholar
• P. Bochev and M. Gunzburger An Absolutely Stable Pressure-Poisson Stabilized Finite Element Method for the Stokes Equations, SIAM J. Numer. Anal., vol. 42, pp. 1189–1207, 2004.
   Web of Science ®Google Scholar
• F. Hecht, O. Pironneau, A. L. Hyaric, and K. Ohtsuka Freefem++, version 2.3-3, 2008. http://www.freefem.org
   Google Scholar