In this paper we consider the flux-free finite element method based on the Eulerian framework for immiscible incompressible two-fluid flows, which is defined so as to preserve the mass of each fluid. This method is derived from the variational formulation including the flux-free constraint for the Navier–Stokes equations by the Lagrange multiplier technique. Focusing on the stationary problem, we prove the well-posedness of the finite element solution by a discrete inf-sup condition and show basic error estimates. Moreover we also show the stability of the fractional-step projection finite element scheme for the non-stationary problem. Finally, we give some numerical results to validate our method.
Similar content being viewed by others
References
Dervieux A., Thomasset F. (1980). A finite element method for the simulation of a Rayleigh-Taylor instability. Lec. Notes Math. 771, 145–158
Formaggia L., Gerbeau J.-F., Nobile F., Quarteroni A. (2002). Numerical of defective boundary conditions for the Navier–Stokes equations. SIAM J. Numer. Anal. 40, 376–401
Gerbeau J.-F., Lelièvre T., Le Bris C. (2003). Simulations of MHD flows with moving interfaces. J. Comput. Phys. 184, 163–191
Girault V., Raviart P.-A. (1986). Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin
Glowinski R., Le Tallec P., Ravachol M., and Tsikkinis V. (1992). Numerical solution of the Navier–Stokes equations modeling the flow of two incompressible nonmiscible viscous fluids. Finite elements in fluids, vol. 8, Academic Press, pp. 137–163
Grisvard P. (1985). Elliptic Problems in Nonsmooth Domains. Pitman, Boston
Lions J.-L., Magenes E. (1972). Non-Homogeneous Boundary Value Problems and Apprications I. Springer-Verlag, Berlin
Maronnier V., Picasso M., Rappaz J. (1999). Numerical simulation of free surface flows. J. Comput. Phys. 155, 439–455
Ohmori K. (2002). Convergence of the interface in the finite element approximation for two-fluid flows. In Salvi R. (ed.), The Navier–Stokes equations: theory and numerical methods, Lecture Notes in Pure and Applied Mathematics, Vol. 223, Marcel Dekker, NY, pp. 279–293
Ohmori K., Fujima S., Fujita Y. (2003). Convergence analysis of the interface for interfacial transport phenomena. Math. J. Toyama Univ. 26, 109–129
Ohmori K., and Okumura H. Numerical simulation of immiscible two-fluid flows by flux-free finite element method, preprint.
Quartapelle L. (1993). Numerical solution of the incompressible Navier–Stokes equations. Birkhäuser Verlag Basel, Boston Berlin
Quarteroni A., Saleri F., Veneziani A. (2000). Factorization methods for the numerical approximation of Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg. 188, 505–526
Quarteroni A., Valli A. (1999). Domain Decomposition Methods for Partial Differential Equations. Clarendon Press, Oxford
Saito N., Fujita H. (2000). Remarks on traces of H 1-functions defined in a domain with corners. J. Math. Sci. Univ. Tokyo 7, 325–345
Sussman M., Smereka P., Osher S. (1994). A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ohmori, K., Saito, N. Flux-free Finite Element Method with Lagrange Multipliers for Two-fluid Flows. J Sci Comput 32, 147–173 (2007). https://doi.org/10.1007/s10915-006-9127-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-006-9127-3