A time dependent Stokes interface problem: well-posedness and space-time finite element discretization

Voulis, Igor; Reusken, Arnold

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2187-2213.

In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of the unfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

[1] H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interface Free Bound. 9 (2007) 31–65. | DOI | MR | Zbl

[2] H. Abels, H. Garcke, Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows. Springer International Publishing, Cham (2016) 1–60. | MR

[3] N. Ahmed, S. Becher, G. Matthies, Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem. Comput. Methods Appl. Mech. Eng. 313 (2017) 28–52. | DOI | MR | Zbl

[4] H.W. Alt, Linear Functional Analysis. Springer London, London (2016). | DOI | MR | Zbl

[5] E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math. 88 (2001) 203–235. | DOI | MR | Zbl

[6] R. Becker, E. Burman, P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198 (2009) 3352–3360. | DOI | MR | Zbl

[7] D. Bothe, A. Reusken, Transport Processes at Fluidic Interfaces, Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel (2017). | DOI | MR | Zbl

[8] E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104 (2015) 472–501. | DOI | MR | Zbl

[9] E. Burman, P. Hansbo, Fictitious domain finite element methods using cut elements: II. a stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328–341. | DOI | MR | Zbl

[10] E. Burman, P. Hansbo, Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM: M2AN 48 (2014) 859–874. | DOI | Numdam | MR | Zbl

[11] Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175–202. | DOI | MR | Zbl

[12] G. Crippa, The Flow Associated to Weakly Diffierentiable Vector Fields. Edizioni della Normale, Pisa (2009). | MR | Zbl

[13] R. Croce, M. Griebel, M.A. Schweitzer, Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions. Int. J. Numer. Methods Fluids 62 (2010) 963–993. | MR | Zbl

[14] I.V. Denisova, V.A. Solonnikov, Classical solvability of the problem of the motion of two viscous incompressible fluids. St. Petersburg Math. J. 7 (1996) 755–786. | MR | Zbl

[15] I.V. Denisova, V.A. Solonnikov, Global solvability of a problem governing the motion of two incompressible capillary fluids in a container. J. Math. Sci. 185 (2012) 668–686. | DOI | MR | Zbl

[16] R.J. Diperna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | DOI | MR | Zbl

[17] A. Ern, J. Guermond, Theory and Practice of Finite Elements. Springer New York, New York, NY (2013). | MR | Zbl

[18] L. Evans, Partial Differential Equations. American Mathematical Society, Providence, RI (2010). | MR | Zbl

[19] R.S. Falk, M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51 (2013) 1308–1326. | DOI | MR | Zbl

[20] T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84 (2010) 253–304. | DOI | MR | Zbl

[21] J. Grande, Finite element discretization error analysis of a general interfacial stress functional. SIAM J. Numer. Anal. 53 (2015) 1236–1255. | DOI | MR | Zbl

[22] S. Groß, A. Reusken, Numerical Methods for Two-phase Incompressible Flows. Springer, Berlin Heidelberg, Berlin (2011). | DOI | MR | Zbl

[23] R. Guberovic, C. Schwab, R. Stevenson, Space-time variational saddle point formulations of Stokes and Navier-Stokes equations. ESAIM: M2AN 48 (2014) 875–894. | DOI | Numdam | MR | Zbl

[24] A. Hansbo, P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (2002) 5537–5552. | DOI | MR | Zbl

[25] P. Hansbo, M.G. Larson, S. Zahedi, A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85 (2014) 90–114. | DOI | MR | Zbl

[26] V. John, A. Linke, C. Merdon, M. Neilan, L. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59 (2017) 492–544. | DOI | MR | Zbl

[27] M. Kirchhart, S. Groß, A. Reusken, Analysis of an XFEM discretization for Stokes interface problems. SIAM J. Sci. Comput. 38 (2016) A1019–A1043. | DOI | MR | Zbl

[28] C. Lehrenfeld, The Nitsche XFEM-DG space-time method and its implementation in three space dimensions. SIAM J. Sci. Comput. 37 (2015) A245–A270. | DOI | MR | Zbl

[29] C. Lehrenfeld, A. Reusken, Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems. SIAM J. Numer. Anal. 51 (2013) 958–983. | DOI | MR | Zbl

[30] A. Lozovskiy, M.A. Olshanskii, Y.V. Vassilevski, A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain. Comput. Methods Appl. Mech. Eng. 333 (2018) 55–73. | DOI | MR | Zbl

[31] A. Nouri, F. Poupaud, An existence theorem for the multifluid Navier-Stokes problem. J. Differ. Equ. 122 (1995) 71–88. | DOI | MR | Zbl

[32] A. Nouri, F. Poupaud, Y. Demay, An existence theorem for the multi-fluid Stokes problem. Q. Appl. Math. 55 (1997) 421–435. | DOI | MR | Zbl

[33] J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 10 (2010) 311–345. | DOI | MR | Zbl

[34] J. Prüuss, G. Simonett, Analytic Solutions for the Two-phase Navier-Stokes Equations with Surface Tension and Gravity. Springer Basel, Basel (2011) 507–540. | MR | Zbl

[35] J. Prüuss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Birkhäuser, Basel (2016). | DOI | MR

[36] J. Saal, Maximal regularity for the Stokes system on noncylindrical space-time domains. J. Math. Soc. Jpn. 58 (2006) 617–641. | DOI | MR | Zbl

[37] J. San Martín, L. Smaranda, T. Takahashi, Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time. J. Comput. Appl. Math. 230 (2009) 521–545. | DOI | MR | Zbl

[38] C. Schwab, R. Stevenson, Fractional space-time variational formulations of (Navier–) Stokes equations. SIAM J. Math. Anal. 49 (2017) 2442–2467. | DOI | MR | Zbl

[39] V.A. Solonnikov, On the problem of non-stationary motion of two viscous incompressible liquids. J. Math. Sci. 142 (2007) 1844–1866. | DOI | MR | Zbl

[40] O. Steinbach, H. Yang, Comparison of algebraic multigrid methods for an adaptive space–time finite-element discretization of the heat equation in 3D and 4D. Numer. Linear Algebra Appl. 25 (2018) e2143. | DOI | MR | Zbl

[41] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam (1977). | MR | Zbl

[42] V. Thoméee, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York Inc., New York (2006). | MR

[43] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR | Zbl