Finite element error analysis of surface Stokes equations in stream function formulation

Brandner, Philip; Reusken, Arnold

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 2069-2097.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ³ without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk–Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability. Bull. Sci. Math. 136 (2012) 857–881. | DOI | MR | Zbl

V.I. Arnol’D, Mathematical Methods of Classical Mechanics. Springer Science & Business Media 60 (2013). | MR | Zbl

M. Arroyo and A. Desimone, Relaxation dynamics of fluid membranes. Phys. Rev. E 79 (2009) 031915. | DOI | MR

I. Babuska, J. Osborn and J. Pitkaranta, Analysis of mixed methods using mesh dependent norms. Math. Comput. 35 (1980) 1039–1062. | DOI | MR | Zbl

J.W. Barrett, H. Garcke and R. Nürnberg, A stable numerical method for the dynamics of fluidic membranes. Numer. Math. 134 (2016) 783–822. | DOI | MR

A. Bonito, A. Demlow and M. Licht, A divergence-conforming finite element method for the surface Stokes equation. Preprint (2019). | arXiv | MR

A. Bonito, A. Demlow and R.H. Nochetto, Finite element methods for the Laplace-Beltrami operator. Preprint (2019). | arXiv | MR

H. Brenner, Interfacial Transport Processes and Rheology. Elsevier (2013).

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

E. Burman, P. Hansbo, M.G. Larson and A. Massing, Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM: M2AN 52 (2018) 2247–2282. | DOI | Numdam | MR

P. Ciarlet and P. Raviart, A mixed finite element method for the biharmonic equation. In: Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, edited by C. De Boor. Academic Press (1974) 125–143. | DOI | MR | Zbl

G. Dziuk, Finite elements for the beltrami operator on arbitrary surfaces. In: Vol. 1357 of Partial Differential Equations and Calculus of Variations, edited by S. Hildebrandt and R. Leis. Lecture Notes in Mathematics. Springer (1988) 142–155. | MR | Zbl

G. Dziuk and C.M. Elliott, Finite element methods for surface PDEs. Acta Numer. 22 (2013) 289–396. | DOI | MR | Zbl

D.G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92 (1970) 102–163. | DOI | MR | Zbl

R.S. Falk and J.E. Osborn, Error estimates for mixed methods. RAIRO: Anal. Numer. 14 (1980) 249–277. | Numdam | MR | Zbl

T.-P. Fries, Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds. Int. J. Numer. Methods Fluids 88 (2018) 55–78. | DOI | MR

V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986). | DOI | MR | Zbl

J. Grande, C. Lehrenfeld and A. Reusken, Analysis of a high-order trace finite element method for PDEs on level set surfaces. SIAM J. Numer. Anal. 56 (2018) 228–255. | DOI | MR

M.E. Gurtin and A.I. Murdoch, A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57 (1975) 291–323. | DOI | MR | Zbl

A. Hansbo and 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

T. Jankuhn, M. Olshanskii and A. Reusken, Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound. 20 (2018) 353–377. | DOI | MR

H. Koba, C. Liu and Y. Giga, Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Math. 75 (2017) 359–389. | DOI | MR

P.L. Lederer, C. Lehrenfeld and J. Schöberl, Divergence-free tangential finite element methods for incompressible flows on surfaces. Preprint (2019). | arXiv | MR

C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings. Comput. Methods Appl. Mech. Eng. 300 (2016) 716–733. | DOI | MR

M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321 (2001) 955–987. | DOI | MR | Zbl

T.-H. Miura, On singular limit equations for incompressible fluids in moving thin domains. Q. Appl. Math. 76 (2017). | MR

Netgen/ngsolve. https://ngsolve.org/ (2020).

ngsxfem. https://github.com/ngsxfem (2020).

I. Nitschke, A. Voigt and J. Wensch, A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708 (2012) 418–438. | DOI | MR | Zbl

M.A. Olshanskii and A. Reusken, Trace finite element methods for PDEs on surfaces. In: Geometrically Unfitted Finite Element Methods and Applications, edited by S.P.A. Bordas, E. Burman, M.G. Larson and M.A. Olshanskii. Springer International Publishing, Cham (2017) 211–258. | DOI | MR

M.A. Olshanskii and V. Yushutin, A penalty finite element method for a fluid system posed on embedded surface. J. Math. Fluid Mech. 21 (2018) 1–18. | MR

M. Olshanskii, A. Quaini, A. Reusken and V. Yushutin, A finite element method for the surface Stokes problem. SIAM J. Sci. Comput. 40 (2018) A2492–A2518. | DOI | MR

M.A. Olshanskii, A. Reusken and A. Zhiliakov, Inf-sup stability of the trace $P_1-P_2$ Taylor–Hood elements for surface PDEs. To appear in: Math. Comp. (2019). Available by: | DOI | MR

P. Petersen, Riemannian Geometry, 3rd edition. Springer International Publishing (2016). | DOI | MR | Zbl

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994). | DOI | MR | Zbl

M. Rahimi, A. Desimone and M. Arroyo, Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matter 9 (2013) 11033–11045. | DOI

P. Rangamani, A. Agrawal, K.K. Mandadapu, G. Oster and D.J. Steigmann, Interaction between surface shape and intra-surface viscous flow on lipid membranes. Biomech. Model. Mechanobiol. 12 (2013) 833–845. | DOI

A. Reusken, Analysis of trace finite element methods for surface partial differential equations. IMA J. Numer. Anal. 35 (2015) 1568–1590. | DOI | MR

A. Reusken, Stream function formulation of surface Stokes equations. IMA J. Numer. Anal. 40 (2018) 109–139. | DOI | MR

S. Reuther and A. Voigt, The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 13 (2015) 632–643. | DOI | MR

S. Reuther and A. Voigt, Solving the incompressible surface Navier-Stokes equation by surface finite elements. Phys. Fluids 30 (2018) 012107. | DOI

T. Sakai, Riemannian Geometry. In vol. 149 of Translations of Mathematical Monographs. American Mathematical Soc. (1996). | MR | Zbl

J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Visual. Sci. 1 (1997) 41–52. | DOI | Zbl

L. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12 (1960) 98–108. | DOI

J.C. Slattery, L. Sagis and E.-S. Oh, Interfacial Transport Phenomena. Springer Science & Business Media (2007). | MR | Zbl

M.E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Commun. Part. Differ. Equ. 17 (1992) 1407–1456. | DOI | MR | Zbl

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988). | DOI | MR | Zbl