An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces
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- by Maxim A. Olshanskii, Arnold Reusken and Paul Schwering;
- Math. Comp. 93 (2024), 2031-2065
- DOI: https://doi.org/10.1090/mcom/3931
- Published electronically: December 11, 2023
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Abstract:
The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier–Stokes equations posed on a passively evolving smooth closed surface embedded in $\mathbb {R}^3$. The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.References
- Netgen/NGSolve, https://ngsolve.org/.
- Andrea Bonito, Alan Demlow, and Martin Licht, A divergence-conforming finite element method for the surface Stokes equation, SIAM J. Numer. Anal. 58 (2020), no. 5, 2764–2798. MR 4155235, DOI 10.1137/19M1284592
- Philip Brandner, Thomas Jankuhn, Simon Praetorius, Arnold Reusken, and Axel Voigt, Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations, SIAM J. Sci. Comput. 44 (2022), no. 4, A1807–A1832. MR 4447436, DOI 10.1137/21M1403126
- Erik Burman, Stefan Frei, and Andre Massing, Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains, Numer. Math. 150 (2022), no. 2, 423–478. MR 4382585, DOI 10.1007/s00211-021-01264-x
- Erik Burman, Peter Hansbo, Mats G. Larson, and André Massing, Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 6, 2247–2282. MR 3905189, DOI 10.1051/m2an/2018038
- P. Cicuta, S. L. Keller, and S. L. Veatch, Diffusion of liquid domains in lipid bilayer membranes, J. Phys. Chem. B 111 (2007), 3328–3331.
- Eloy de Kinkelder, Leonard Sagis, and Sebastian Aland, A numerical method for the simulation of viscoelastic fluid surfaces, J. Comput. Phys. 440 (2021), Paper No. 110413, 18. MR 4262048, DOI 10.1016/j.jcp.2021.110413
- R. Dimova, S. Aranda, N. Bezlyepkina, V. Nikolov, K. A. Riske, and R. Lipowsky, A practical guide to giant vesicles. Probing the membrane nanoregime via optical microscopy, J. Phys. Condens. Matter 18 (2006), S1151.
- Gerhard Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial differential equations and calculus of variations, Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988, pp. 142–155. MR 976234, DOI 10.1007/BFb0082865
- Thomas-Peter Fries, Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds, Internat. J. Numer. Methods Fluids 88 (2018), no. 2, 55–78. MR 3846120, DOI 10.1002/fld.4510
- Thomas-Peter Fries and Samir Omerović, Higher-order accurate integration of implicit geometries, Internat. J. Numer. Methods Engrg. 106 (2016), no. 5, 323–371. MR 3491699, DOI 10.1002/nme.5121
- Jörg Grande, Christoph Lehrenfeld, and Arnold Reusken, Analysis of a high-order trace finite element method for PDEs on level set surfaces, SIAM J. Numer. Anal. 56 (2018), no. 1, 228–255. MR 3747467, DOI 10.1137/16M1102203
- B. J. Gross, N. Trask, P. Kuberry, and P. J. Atzberger, Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: a Generalized Moving Least-Squares (GMLS) approach, J. Comput. Phys. 409 (2020), 109340, 23. MR 4074520, DOI 10.1016/j.jcp.2020.109340
- Sven Gross, Thomas Jankuhn, Maxim A. Olshanskii, and Arnold Reusken, A trace finite element method for vector-Laplacians on surfaces, SIAM J. Numer. Anal. 56 (2018), no. 4, 2406–2429. MR 3840893, DOI 10.1137/17M1146038
- Morton E. Gurtin and A. Ian Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57 (1975), 291–323. MR 371223, DOI 10.1007/BF00261375
- Peter Hansbo, Mats G. Larson, and Karl Larsson, Analysis of finite element methods for vector Laplacians on surfaces, IMA J. Numer. Anal. 40 (2020), no. 3, 1652–1701. MR 4122487, DOI 10.1093/imanum/drz018
- Thomas Jankuhn, Maxim A. Olshanskii, and Arnold Reusken, Incompressible fluid problems on embedded surfaces: modeling and variational formulations, Interfaces Free Bound. 20 (2018), no. 3, 353–377. MR 3875687, DOI 10.4171/IFB/405
- Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken, and Alexander Zhiliakov, Error analysis of higher order trace finite element methods for the surface Stokes equation, J. Numer. Math. 29 (2021), no. 3, 245–267. MR 4317295, DOI 10.1515/jnma-2020-0017
- T. Jankuhn and A. Reusken, Higher order trace finite element methods for the surface Stokes equation, Preprint, arXiv:1909.08327, (2019).
- T. Jankuhn and A. Reusken, Trace finite element methods for surface vector-Laplace equations, IMA J. Numer. Anal. 41 (2020), 48–83.
- F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Topology and dynamics of active nematic vesicles, Science 345 (2014), 1135–1139.
- Hajime Koba, Chun Liu, and Yoshikazu Giga, Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no. 2, 359–389. MR 3614501, DOI 10.1090/qam/1452
- Philip L. Lederer, Christoph Lehrenfeld, and Joachim Schöberl, Divergence-free tangential finite element methods for incompressible flows on surfaces, Internat. J. Numer. Methods Engrg. 121 (2020), no. 11, 2503–2533. MR 4156753, DOI 10.1002/nme.6317
- Christoph Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg. 300 (2016), 716–733. MR 3452792, DOI 10.1016/j.cma.2015.12.005
- C. Lehrenfeld, F. Heimann, J. Preuß, and H. von Wahl, ‘ngsxfem’: add-on to ngsolve for geometrically unfitted finite element discretizations, J. Open Source Softw. 6 (2021), 3237.
- Christoph Lehrenfeld, Maxim A. Olshanskii, and Xianmin Xu, A stabilized trace finite element method for partial differential equations on evolving surfaces, SIAM J. Numer. Anal. 56 (2018), no. 3, 1643–1672. MR 3816183, DOI 10.1137/17M1148633
- Maplesoft, Maple, https://de.maplesoft.com/.
- Tatsu-Hiko Miura, On singular limit equations for incompressible fluids in moving thin domains, Quart. Appl. Math. 76 (2018), no. 2, 215–251. MR 3769895, DOI 10.1090/qam/1495
- B. Müller, F. Kummer, and M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Internat. J. Numer. Methods Engrg. 96 (2013), no. 8, 512–528. MR 3130061, DOI 10.1002/nme.4569
- A. I. Murdoch and H. Cohen, Symmetry considerations for material surfaces, Arch. Rational Mech. Anal. 72 (1979/80), no. 1, 61–98. MR 540222, DOI 10.1007/BF00250737
- I. Nitschke, S. Reuther, and A. Voigt, Hydrodynamic interactions in polar liquid crystals on evolving surfaces, Phys. Rev. Fluids 4 (2019), 044002.
- Maxim A. Olshanskii, Arnold Reusken, and Jörg Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3339–3358. MR 2551197, DOI 10.1137/080717602
- Maxim A. Olshanskii, Arnold Reusken, and Alexander Zhiliakov, Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor-Hood elements for surface PDEs, Math. Comp. 90 (2021), no. 330, 1527–1555. MR 4273108, DOI 10.1090/mcom/3551
- Maxim A. Olshanskii, Annalisa Quaini, Arnold Reusken, and Vladimir Yushutin, A finite element method for the surface Stokes problem, SIAM J. Sci. Comput. 40 (2018), no. 4, A2492–A2518. MR 3841618, DOI 10.1137/18M1166183
- Maxim A. Olshanskii and Arnold Reusken, Trace finite element methods for PDEs on surfaces, Geometrically unfitted finite element methods and applications, Lect. Notes Comput. Sci. Eng., vol. 121, Springer, Cham, 2017, pp. 211–258. MR 3806652, DOI 10.1007/978-3-319-71431-8_{7}
- Maxim A. Olshanskii, Arnold Reusken, and Xianmin Xu, An Eulerian space-time finite element method for diffusion problems on evolving surfaces, SIAM J. Numer. Anal. 52 (2014), no. 3, 1354–1377. MR 3215065, DOI 10.1137/130918149
- Maxim A. Olshanskii, Arnold Reusken, and Alexander Zhiliakov, Tangential Navier-Stokes equations on evolving surfaces: analysis and simulations, Math. Models Methods Appl. Sci. 32 (2022), no. 14, 2817–2852. MR 4546904, DOI 10.1142/S0218202522500658
- M. A. Olshanskii and D. Safin, Numerical integration over implicitly defined domains for higher order unfitted finite element methods, Lobachevskii J. Math. 37 (2016), no. 5, 582–596. MR 3549486, DOI 10.1134/S1995080216050103
- Maxim A. Olshanskii and Vladimir Yushutin, A penalty finite element method for a fluid system posed on embedded surface, J. Math. Fluid Mech. 21 (2019), no. 1, Paper No. 14, 18. MR 3911729, DOI 10.1007/s00021-019-0420-y
- M. Rank and A. Voigt, Active flows on curved surfaces, Phys. Fluids 33 (2021), 072110.
- Arnold Reusken, Analysis of trace finite element methods for surface partial differential equations, IMA J. Numer. Anal. 35 (2015), no. 4, 1568–1590. MR 3407236, DOI 10.1093/imanum/dru047
- Arnold Reusken, Stream function formulation of surface Stokes equations, IMA J. Numer. Anal. 40 (2020), no. 1, 109–139. MR 4050536, DOI 10.1093/imanum/dry062
- S. Reuther and A. Voigt, The interplay of curvature and vortices in flow on curved surfaces, Multiscale Model. Simul. 13 (2015), no. 2, 632–643. MR 3359666, DOI 10.1137/140971798
- S. Reuther and A. Voigt, Solving the incompressible surface Navier–Stokes equation by surface finite elements, Phys. Fluids 30 (2018), 012107.
- R. I. Saye, High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput. 37 (2015), no. 2, A993–A1019. MR 3338676, DOI 10.1137/140966290
- Pratik Suchde, A meshfree Lagrangian method for flow on manifolds, Internat. J. Numer. Methods Fluids 93 (2021), no. 6, 1871–1894. MR 4252630, DOI 10.1002/fld.4957
- Y. Sudhakar and Wolfgang A. Wall, Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods, Comput. Methods Appl. Mech. Engrg. 258 (2013), 39–54. MR 3047898, DOI 10.1016/j.cma.2013.01.007
- Ming Sun, Xufeng Xiao, Xinlong Feng, and Kun Wang, Modeling and numerical simulation of surfactant systems with incompressible fluid flows on surfaces, Comput. Methods Appl. Mech. Engrg. 390 (2022), Paper No. 114450, 24. MR 4357312, DOI 10.1016/j.cma.2021.114450
- Henry von Wahl, Thomas Richter, and Christoph Lehrenfeld, An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains, IMA J. Numer. Anal. 42 (2022), no. 3, 2505–2544. MR 4454929, DOI 10.1093/imanum/drab044
- Arash Yavari, Arkadas Ozakin, and Souhayl Sadik, Nonlinear elasticity in a deforming ambient space, J. Nonlinear Sci. 26 (2016), no. 6, 1651–1692. MR 3562390, DOI 10.1007/s00332-016-9315-8
Bibliographic Information
- Maxim A. Olshanskii
- Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
- MR Author ID: 343398
- ORCID: 0000-0002-9102-6833
- Email: maolshanskiy@uh.edu
- Arnold Reusken
- Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D-52056 Aachen, Germany
- MR Author ID: 147305
- ORCID: 0000-0002-4713-9638
- Email: reusken@igpm.rwth-aachen.de
- Paul Schwering
- Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D-52056 Aachen, Germany
- MR Author ID: 1540111
- Email: schwering@igpm.rwth-aachen.de
- Received by editor(s): February 1, 2023
- Received by editor(s) in revised form: October 13, 2023
- Published electronically: December 11, 2023
- Additional Notes: The second and third authors were financially supported by the German Research Foundation (DFG) within the Research Unit “Vector- and tensor valued surface PDEs” (FOR 3013) with project no. RE 1461/11-2. The first author was partially supported by US National Science Foundation (NSF) through DMS-2309197.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2031-2065
- MSC (2020): Primary 65M12, 65M15, 65M60
- DOI: https://doi.org/10.1090/mcom/3931
- MathSciNet review: 4759369