Abstract
We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier–Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an \(H^1\)-conforming finite element method, and an upwind discontinuous Galerkin finite element method with post-processed velocity is adopted for the density equation. The proposed method is proved to be convergent in approximating reasonably smooth solutions in three-dimensional convex polyhedral domains.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of Data and Materials
Not applicable.
Code Availability Statement
Not applicable.
References
Adams, R., Fournier, J.: Sobolev Spaces, Pure and Applied Mathematics. Elsevier Science, Amsterdam (2003)
Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 1–46 (1998)
An, R.: Error analysis of a new fractional-step method for the incompressible Navier–Stokes equations with variable density. J. Sci. Comput. (2020). https://doi.org/10.1007/s10915-020-01253-6
Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the stokes equations. Calcolo 21(4), 337–344 (1984)
Bell, J.B., Marcus, D.L.: A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348 (1992)
Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed Finite Elements, Compatibility Conditions, and Applications. Springer, Berlin (2008)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)
Cai, W., Li, B., Li, Y.: Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions. ESAIM: Math. Model. Numer. Anal. (2020). https://doi.org/10.1051/m2an/2020029
Danchin, R.: Density-dependent incompressible fluids in bounded domains. J. Math. Fluid Mech. 8, 333–381 (2006)
Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin (2012)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Society for Industrial and Applied Mathematics, Philadelphia (2011)
Guermond, J.L., Quartapelle, L.: A projection FEM for variable density incompressible flows. J. Comput. Phys. 165, 167–188 (2000)
Guermond, J.L., Salgado, A.: A fractional step method based on a pressure Poisson equation for incompressible flows with variable density. C. R. Math. 346, 913–918 (2008)
Guermond, J.L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228, 2834–2846 (2009)
Guermond, J.L., Salgado, A.: Error analysis of a fractional time-stepping technique for incompressible flows with variable density. SIAM J. Numer. Anal. 49, 917–940 (2011)
Ladyzhenskaya, O., Solonnikov, V.: Unique solvability of an initial- and boundary-value problem for viscous incompressible inhomogeneous fluids. J. Sov. Math. 9, 697–749 (1978)
Latché, J.C., Saleh, K.: A convergent staggered scheme for the variable density incompressible Navier–Stokes equations. Math. Comput. 87, 581–632 (2018)
Li, Y., Li, J., Mei, L., Li, Y.: Mixed stabilized finite element methods based on backward difference/Adams–Bashforth scheme for the time-dependent variable density incompressible flows. Comput. Math. Appl. 70, 2575–2588 (2015)
Li, Y., Mei, L., Ge, J., Shi, F.: A new fractional time-stepping method for variable density incompressible flows. J. Comput. Phys. 242, 124–137 (2013)
Liu, C., Walkington, N.J.: Convergence of numerical approximations of the incompressible Navier–Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45, 1287–1304 (2007)
Ortega-Torres, E., Braz e Silva, P., Rojas-Medar, M.: Analysis of an iterative method for variable density incompressible fluids. Annali Dell’universita Di Ferrara’ 55, 129 (2009)
Pyo, J.-H., Shen, J.: Gauge–Uzawa methods for incompressible flows with variable density. J. Comput. Phys. 221, 181–197 (2007)
Rathgeber, F., Ham, D.A., Mitchell, L., Lange, M., Luporini, F., McRae, A.T.T., Bercea, G.-T., Markall, G.R., Kelly, P.H.J.: Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. 43, 1–27 (2016)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, New York (2006)
Funding
The work of B. Li and Z. Yang was supported in part by National Natural Science Foundation of China (NSFC Grant 12071020) and an internal grant of The Hong Kong Polytechnic University (Project 4-ZZKQ). Weifeng Qiu is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302718).
Author information
Authors and Affiliations
Contributions
BL, WQ and ZY have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, method, analysis and writing. All authors certify that this material or similar material has not been and will not be submitted to or published in any other publication.
Corresponding author
Ethics declarations
Conflict of interest
No conflict of interest exists.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, B., Qiu, W. & Yang, Z. A Convergent Post-processed Discontinuous Galerkin Method for Incompressible Flow with Variable Density. J Sci Comput 91, 2 (2022). https://doi.org/10.1007/s10915-022-01775-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01775-1