Error analysis of higher order Trace Finite Element Methods for the surface Stokes equation

References

[1] M. Arroyo and A. DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E 79 (2009), No. 3, 031915.10.1103/PhysRevE.79.031915Suche in Google Scholar PubMed

[2] A. Bonito, A. Demlow, and M. Licht, A divergence-conforming finite element method for the surface Stokes equation, SIAM J. Numer. Anal., 58 (2020), No. 5, 2764–2798.10.1137/19M1284592Suche in Google Scholar

[3] P. Brandner and A. Reusken, Finite element error analysis of surface Stokes equations in stream function formulation, ESAIM: M2AN, 54 (2020), No. 6, 2069–2097.10.1051/m2an/2020044Suche in Google Scholar

[4] H. Brenner, Interfacial Transport Processes and Rheology, Elsevier, 2013.Suche in Google Scholar

[5] E. Burman and A. Ern, An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal. 56 (2018), 1525–1546.10.1137/17M1154266Suche in Google Scholar

[6] 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: Math. Modelling Numer. Analysis 52 (2018), No. 6, 2247–2282.10.1051/m2an/2018038Suche in Google Scholar

[7] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci Flow: Techniques and Applications. Part IV: Long-Time Solutions and Related Topics, American Mathematical Society, 2007.10.1090/surv/144Suche in Google Scholar

[8] A. Demlow and G. Dziuk, An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45 (2007), 421–442.10.1137/050642873Suche in Google Scholar

[9] DROPS package, http://www.igpm.rwth-aachen.de/DROPS/.Suche in Google Scholar

[10] G. Dziuk, Finite elements for the Beltrami Operator on arbitrary surfaces, In: Partial Differential Equations and Calculus of Variations (Eds. S. Hildebrandt and R. Leis), Lecture Notes in Mathematics, Vol. 1357, Springer, 1988, pp. 142–155.10.1007/BFb0082865Suche in Google Scholar

[11] T.-P. Fries, Higher-order surface FEM for incompressible Navier–Stokes flows on manifolds, Int. J. Numer. Methods Fluids 88 (2018), No. 2, 55–78.10.1002/fld.4510Suche in Google Scholar

[12] G. G. Fuller and J. Vermant, Complex fluid–fluid interfaces: rheology and structure, Annual Review of Chemical and Biomolecular Engrg. 3 (2012), 519–543.10.1146/annurev-chembioeng-061010-114202Suche in Google Scholar PubMed

[13] 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. Analysis 56 (2018), No. 1, 228–255.10.1137/16M1102203Suche in Google Scholar

[14] 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. Comp. Phys., 409 (2020), 109340.10.1016/j.jcp.2020.109340Suche in Google Scholar

[15] M. E. Gurtin and A. I. Murdoch, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57 (1975), No. 4, 291–323.10.1007/BF00261375Suche in Google Scholar

[16] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), 5537–5552.10.1016/S0045-7825(02)00524-8Suche in Google Scholar

[17] P. Hansbo, M. G. Larson, and K. Larsson, Analysis of finite element methods for vector Laplacians on surfaces, IMA J. Numer. Anal. 40 (2020), No, 3, 1652–1701.10.1093/imanum/drz018Suche in Google Scholar

[18] T. Heister and G. Rapin, Efficient augmented Lagrangian-type preconditioning for the Oseen problem using Grad-Div stabilization, Int. J. Numer. Methods Fluids 71 (2013), No. 1, 118–134.10.1002/fld.3654Suche in Google Scholar

[19] T. Jankuhn, M. A. Olshanskii, and A. Reusken, Incompressible fluid problems on embedded surfaces: Modeling and variational formulations, Interfaces and Free Boundaries 20 (2018), 353–377.10.4171/IFB/405Suche in Google Scholar

[20] T. Jankuhn and A. Reusken, Higher order trace finite element methods for the surface Stokes equation, Preprint arXiv:1909.08327, 2019.Suche in Google Scholar

[21] T. Jankuhn and A. Reusken, Trace finite element methods for surface vector-Laplace equations, IMA J. Numer. Anal. 41 (2021), No. 1, 48–83.10.1093/imanum/drz062Suche in Google Scholar

[22] P. L. Lederer, C. Lehrenfeld, and J. Schöberl, Divergence-free tangential finite element methods for incompressible flows on surfaces, Int. J. Numer. Meth. Engrg., 121 (2020), No. 11, 2503–2533.10.1002/nme.6317Suche in Google Scholar PubMed PubMed Central

[23] C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comp. Methods Appl. Mechanics Engrg. 300 (2016), 716–733.10.1016/j.cma.2015.12.005Suche in Google Scholar

[24] C. Lehrenfeld and A. Reusken, Analysis of a high-order unfitted finite element method for elliptic interface problems, IMA J. Numer. Anal. 38 (2017), No. 3, 1351–1387.10.1093/imanum/drx041Suche in Google Scholar

[25] Netgen/NGSolve, https://ngsolve.org/.Suche in Google Scholar

[26] I. Nitschke, A. Voigt, and J. Wensch, A finite element approach to incompressible two-phase flow on manifolds, J. Fluid Mech. 708 (2012), 418–438.10.1017/jfm.2012.317Suche in Google Scholar

[27] I. Nitschke, S. Reuther, and A. Voigt, Hydrodynamic interactions in polar liquid crystals on evolving surfaces, Phys. Review Fluids 4 (2019), No. 4, 044002.10.1103/PhysRevFluids.4.044002Suche in Google Scholar

[28] M. A. Olshanskii, A. Quaini, A. Reusken, and V. Yushutin, A finite element method for the surface Stokes problem, SIAM J. Sci. Comp. 40 (2018), No. 4, A2492–A2518.10.1137/18M1166183Suche in Google Scholar

[29] M. A. Olshanskii, A. Reusken, and X. Xu, A stabilized finite element method for advection–diffusion equations on surfaces, IMA J. Numer. Anal. 34 (2014), 732–758.10.1093/imanum/drt016Suche in Google Scholar

[30] M. A. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations, Math. Comp. 73 (2004), No. 248, 1699–1718.10.1090/S0025-5718-03-01629-6Suche in Google Scholar

[31] M. A. Olshanskii and A. Reusken, Trace finite element methods for PDEs on surfaces, In: Geometrically Unfitted Finite Element Methods and Applications (Eds. S. P. A. Bordas, E. Burman, M. G. Larson, and M. A. Olshanskii), Springer, Cham, 2017, pp. 211–258.10.1007/978-3-319-71431-8_7Suche in Google Scholar

[32] M. A. Olshanskii, A. Reusken, and J. Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), 3339–3358.10.1137/080717602Suche in Google Scholar

[33] M. A. Olshanskii, A. Reusken, and A. Zhiliakov, Inf-sup stability of the trace P₂-P₁ Taylor–Hood elements for surface PDEs, Math. Comp., 90 (2021), 1527–1555.10.1090/mcom/3551Suche in Google Scholar

[34] M. A. Olshanskii and V. Yushutin, A penalty finite element method for a fluid system posed on embedded surface, J. Math. Fluid Mech. 21 (2019), No. 1, 14.10.1007/s00021-019-0420-ySuche in Google Scholar

[35] M. Rahimi, A. DeSimone, and M. Arroyo, Curved fluid membranes behave laterally as effective viscoelastic media, Soft Matter 9 (2013), No. 46, 11033–11045.10.1039/c3sm51748aSuche in Google Scholar

[36] 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, Biomechanics and Modeling in Mechanobiology 12 (2013), No. 4, 833–845.10.1007/s10237-012-0447-ySuche in Google Scholar PubMed PubMed Central

[37] A. Reusken, Analysis of trace finite element methods for surface partial differential equations, IMA J. Numer. Anal. 35 (2015), No. 4, 1568–1590.10.1093/imanum/dru047Suche in Google Scholar

[38] A. Reusken, Stream function formulation of surface Stokes equations, IMA J. Numer. Anal. 40 (2020), No. 1, 109–139.10.1093/imanum/dry062Suche in Google Scholar

[39] S. Reuther and A. Voigt, Solving the incompressible surface Navier–Stokes equation by surface finite elements, Physics of Fluids 30 (2018), No. 1, 012107.10.1063/1.5005142Suche in Google Scholar

[40] A. Sahu, Y. Omar, R. Sauer, and K. Mandadapu, Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces, J. Comp. Phys. 407 (2020), 109253.10.1016/j.jcp.2020.109253Suche in Google Scholar

[41] P. W. Schroeder, V. John, P. L. Lederer, C. Lehrenfeld, G. Lube, and J. Schöberl, On reference solutions and the sensitivity of the 2D Kelvin–Helmholtz instability problem, Computers & Mathematics Applications 77 (2019), No. 4, 1010–1028.10.1016/j.camwa.2018.10.030Suche in Google Scholar

[42] SciPy, https://www.scipy.org/.Suche in Google Scholar

[43] L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids, Chemical Engrg. Sci. 12 (1960), No. 2, 98–108.10.1016/0009-2509(60)87003-0Suche in Google Scholar

[44] J. C. Slattery, L. Sagis, and E.-S. Oh, Interfacial Transport Phenomena, Springer Science & Business Media, 2007.Suche in Google Scholar

[45] A. Torres-Sanchez, D. Santos-Olivan, and M. Arroyo, Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations, J. Comp. Phys., 405 (2020), No. 1, 109168.10.1016/j.jcp.2019.109168Suche in Google Scholar

[46] A. Torres-Sánchez, D. Millán, and M. Arroyo, Modelling fluid deformable surfaces with an emphasis on biological interfaces, J. Fluid Mech. 872 (2019), 218–271.10.1017/jfm.2019.341Suche in Google Scholar