FINITE ELEMENT ANALYSIS OF LINEARLY EXTRAPOLATED BLENDED BACKWARD DIFFERENCE FORMULA (BLEBDF) FOR THE NATURAL CONVECTION FLOWS

Year 2024, Volume: 6 Issue: 2, 61 - 77, 08.11.2024

Merve Ak Mine Akbas

https://doi.org/10.47087/mjm.1477504

Abstract

In this paper, we study the stability and convergence of fully discrete finite element method with grad-div stabilization for the incompressible non-isothermal fluid flows. The proposed scheme uses finite element discretization in space and linearly extrapolated blended Backward Differentiation Formula (BLEBDF) in time. We prove the unconditional stability over finite time interval and optimally convergence of the scheme. We also present numerical experiments to verify our theoretical convergence rates and show the reliability of the scheme.

Keywords

Finite Element Method, Grad-div stabilization, Natural Convection

References

• R. A. Adams, Sobolev spaces, Academic Press, New York, (1975).
• A. Cıbık, F. G. Eroglu and S. Kaya, Long Time Stability of a Linearly Extrapolated Blended BDF Scheme for Multiphysics Flows, Int. J. Numer. Anal. Mod. 17 (2020) 24-41.
• L. P. Franca and T. J. R. Hughes, Two classes of mixed  nite element methods, Comput. Methods Appl. Mech. Engrg. 69(1), (1988) 89-128.
• J. de Frutos, B. Garcia-Archilla, V. John and J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable  nite elements, Adv Comput. Math. 44, (2018), 195-225.
• K. Galvin, A. Linke, L. Rebholz and N. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg 237-240 (2012) 166-176.
• P. M. Gresho, M. Lee, S. T. Chan and R. L. Sani, Solution of time dependent, incompressible Navier-Stokes and Boussinesq equations using the Galerkin  nite element method, in: Lecture Notes in Math., vol. 771, Springer-Verlag, Berlin/Heidelberg/New York (1980) 203-222.
• V. Girault and P. A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Math., Vol. 749, Springer-Verlag, Berlin (1979).
• E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Sti  and Differential Algebraic Problems, Second edition, Springer-Verlag, Berlin (2002).
• F. Hecht. New development in freefem++, J. Numer. Math., 20(3-4) (2012) 251-265.
• N. Jiang, A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations, Numer. Methods Partial Differ. Equ. 33(1), (2017).
• V. John, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, Switzerland (2016).
• V. John, A. Linke, C. Merdon, M. Neilan and L. Rebholz, On the divergence constraint in mixed  nite element methods for incompressible flows, SIAM Rev. 59 (2017) 492-544.
• W. Layton. An Introduction to the Numerical Analysis of Viscous Incompressible Flows. SIAM, Philadelphia (2008).
• A. Linke, G. Matthies and L. Tobiska, Robust arbitrary order mixed  nite element methods for the incompressible Stokes Equations with pressure independent velocity errors, ESAIM:M2AN. 50 (2016) 289-309.
• A. Linke and C. Merdon, On velocity errors due to irrotational forces in the Navier-Stokes momentum balance, J. Comput. Phys. 313 (2016) 654-661.
• S. Liu, P. Huang and Y. He, A second-order scheme based on blended BDF for the incompressible MHD system, Adv. Comput. Math. 49 (2023) 79.
• J.-G. Liu, C. Wang and H. Johnston Fourth order scheme for incompressible Boussinesq equations, J. Sci. Comput., 18(2) (2003) 253-285.
• H. Melhem, Finite element approximation to heat transfer through combined solid and fluid media, PhD thesis, University of Pittsburgh, (1987).
• M. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations, Math. Comp. 73 (2004) 1699-1718.
• M. A. Olshanskii, G. Lube, T. Heister and J. Löve, Grad-div stabilization and sub-grid pressure models for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 198 (2009) 3975-3988.
• P. H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Benard problem, Arch. Ration. Mech. Anal. 29, (1968).
• S. S. Ravindran, An Analysis of the Blended Three-Step Backward Differentiation Formula Time-Stepping Scheme for the Navier-Stokes-Type System Related to Soret Convection, Numer. Func. Anal. Opt. 36 (2015) 658-686.