A Conservative Eulerian Finite Element Method for Transport and Diffusion in Moving Domains

References

[1] S. Agmon, Lectures on Elliptic Boundary Value Problems, American Mathematical Society, Providence, 1965. Search in Google Scholar

[2] A. Alphonse, C. M. Elliott and B. Stinner, An abstract framework for parabolic PDEs on evolving spaces, Port. Math. 72 (2015), no. 1, 1–46. 10.4171/pm/1955Search in Google Scholar

[3] M. Anselmann and M. Bause, Cut finite element methods and ghost stabilization techniques for space-time discretizations of the Navier–Stokes equations, Internat. J. Numer. Methods Fluids 94 (2022), no. 7, 775–802. 10.1002/fld.5074Search in Google Scholar

[4] S. Badia, E. Neiva and F. Verdugo, Linking ghost penalty and aggregated unfitted methods, Comput. Methods Appl. Mech. Engrg. 388 (2022), Paper No. 114232. 10.1016/j.cma.2021.114232Search in Google Scholar

[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[6] E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris 348 (2010), no. 21–22, 1217–1220. 10.1016/j.crma.2010.10.006Search in Google Scholar

[7] E. Burman, S. Claus, P. Hansbo, M. G. Larson and A. Massing, CutFEM: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg. 104 (2015), no. 7, 472–501. 10.1002/nme.4823Search in Google Scholar

[8] E. Burman, S. Frei and A. Massing, Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains, Numer. Math. 150 (2022), no. 2, 423–478. 10.1007/s00211-021-01264-xSearch in Google Scholar

[9] E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math. 62 (2012), no. 4, 328–341. 10.1016/j.apnum.2011.01.008Search in Google Scholar

[10] S. Claus and P. Kerfriden, A CutFEM method for two-phase flow problems, Comput. Methods Appl. Mech. Engrg. 348 (2019), 185–206. 10.1016/j.cma.2019.01.009Search in Google Scholar

[11] R. Codina, G. Houzeaux, H. Coppola-Owen and J. Baiges, The fixed-mesh ALE approach for the numerical approximation of flows in moving domains, J. Comput. Phys. 228 (2009), no. 5, 1591–1611. 10.1016/j.jcp.2008.11.004Search in Google Scholar

[12] G. Dziuk and C. M. Elliott, L 2 -estimates for the evolving surface finite element method, Math. Comp. 82 (2013), no. 281, 1–24. 10.1090/S0025-5718-2012-02601-9Search in Google Scholar

[13] T. Frachon, E. Nilsson and S. Zahedi, Divergence-free cut finite element methods for Stokes flow, BIT 64 (2024), no. 4, Paper No. 39. 10.1007/s10543-024-01040-xSearch in Google Scholar

[14] S. Frei and T. Richter, A second order time-stepping scheme for parabolic interface problems with moving interfaces, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1539–1560. 10.1051/m2an/2016072Search in Google Scholar

[15] S. Frei and M. K. Singh, An implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain, J. Sci. Comput. 99 (2024), no. 3, Paper No. 64. 10.1007/s10915-024-02530-4Search in Google Scholar

[16] T.-P. Fries and T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications, Internat. J. Numer. Methods Engrg. 84 (2010), no. 3, 253–304. 10.1002/nme.2914Search in Google Scholar

[17] T.-P. Fries and S. Omerović, Higher-order accurate integration of implicit geometries, Internat. J. Numer. Methods Engrg. 106 (2016), no. 5, 323–371. 10.1002/nme.5121Search in Google Scholar

[18] H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci. 28 (2018), no. 3, 525–577. 10.1142/S0218202518500148Search in Google Scholar

[19] P. Hansbo, M. G. Larson and S. Zahedi, A cut finite element method for coupled bulk-surface problems on time-dependent domains, Comput. Methods Appl. Mech. Engrg. 307 (2016), 96–116. 10.1016/j.cma.2016.04.012Search in Google Scholar

[20] F. Hecht and O. Pironneau, An energy stable monolithic Eulerian fluid-structure finite element method, Internat. J. Numer. Methods Fluids 85 (2017), no. 7, 430–446. 10.1002/fld.4388Search in Google Scholar

[21] F. Heimann, C. Lehrenfeld and J. Preuß, Geometrically higher order unfitted space-time methods for PDEs on moving domains, SIAM J. Sci. Comput. 45 (2023), no. 2, B139–B165. 10.1137/22M1476034Search in Google Scholar

[22] F. Heimann, C. Lehrenfeld, P. Stocker and H. von Wahl, Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems, ESAIM Math. Model. Numer. Anal. 57 (2023), no. 5, 2803–2833. 10.1051/m2an/2023064Search in Google Scholar

[23] J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. 10.1137/0727022Search in Google Scholar

[24] C. W. Hirt, A. A. Amsden and J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 14 (1974), no. 3, 227–253. 10.1016/0021-9991(74)90051-5Search in Google Scholar

[25] G. Judakova and M. Bause, Numerical investigation of multiphase flow inpipelines, Internat. Journal. Mech. Mechatron. Engrg. 11 (2017), no. 9, 1540–1546. Search in Google Scholar

[26] C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg. 300 (2016), 716–733. 10.1016/j.cma.2015.12.005Search in Google Scholar

[27] 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), no. 64, Paper No. 3237. 10.21105/joss.03237Search in Google Scholar

[28] C. Lehrenfeld and M. Olshanskii, An Eulerian finite element method for PDEs in time-dependent domains, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 2, 585–614. 10.1051/m2an/2018068Search in Google Scholar

[29] C. Lehrenfeld and A. Reusken, Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems, SIAM J. Numer. Anal. 51 (2013), no. 2, 958–983. 10.1137/120875260Search in Google Scholar

[30] Y. Lou and C. Lehrenfeld, Isoparametric unfitted BDF-finite element method for PDEs on evolving domains, SIAM J. Numer. Anal. 60 (2022), no. 4, 2069–2098. 10.1137/21M142126XSearch in Google Scholar

[31] A. Massing, M. G. Larson, A. Logg and M. E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput. 61 (2014), no. 3, 604–628. 10.1007/s10915-014-9838-9Search in Google Scholar

[32] A. Masud and T. J. R. Hughes, A space-time Galerkin/least-squares finite element formulation of the Navier–Stokes equations for moving domain problems, Comput. Methods Appl. Mech. Engrg. 146 (1997), no. 1–2, 91–126. 10.1016/S0045-7825(96)01222-4Search in Google Scholar

[33] 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. 10.1002/nme.4569Search in Google Scholar

[34] M. Neilan and M. Olshanskii, An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain, IMA J. Numer. Anal. 44 (2024), no. 6, 3234–3258. 10.1093/imanum/drad105Search in Google Scholar

[35] M. A. Olshanskii, A. Reusken and J. Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3339–3358. 10.1137/080717602Search in Google Scholar

[36] M. A. Olshanskii, A. Reusken and X. Xu, An Eulerian space-time finite element method for diffusion problems on evolving surfaces, SIAM J. Numer. Anal. 52 (2014), no. 3, 1354–1377. 10.1137/130918149Search in Google Scholar

[37] 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. 10.1134/S1995080216050103Search in Google Scholar

[38] J. Parvizian, A. Düster and E. Rank, Finite cell method: h- and p-extension for embedded domain problems in solid mechanics, Comput. Mech. 41 (2007), no. 1, 121–133. 10.1007/s00466-007-0173-ySearch in Google Scholar

[39] F. Porté-Agel, M. Bastankhah and S. Shamsoddin, Wind-turbine and wind-farm flows: A review, Bound.-Layer Meteorol. 174 (2019), no. 1, 1–59. 10.1007/s10546-019-00473-0Search in Google Scholar PubMed PubMed Central

[40] J. Preuß, Higher order unfitted isoparametric space-time FEM on moving domains, Master’s thesis, 2018. Search in Google Scholar

[41] 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. 10.1137/140966290Search in Google Scholar

[42] J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci. 1 (1997), no. 1, 41–52. 10.1007/s007910050004Search in Google Scholar

[43] J. Schöberl, C++11 implementation of finite elements in NGSolve, Technical report, Institute for Analysis and Scientific Computing, TU Wien, Wien, 2014. Search in Google Scholar

[44] B. Schott, Stabilized cut finite element methods for complex interface coupled flow problems, PhD thesis, 2017. Search in Google Scholar

[45] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[46] S. Sundaresan, Instabilities in fluidized beds, Annu. Rev. Fluid Mech. 35 (2003), no. 1, 63–88. 10.1146/annurev.fluid.35.101101.161151Search in Google Scholar

[47] T. E. Tezduyar, M. Behr, S. Mittal and J. Liou, A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders, Comput. Methods Appl. Mech. Engrg. 94 (1992), no. 3, 353–371. 10.1016/0045-7825(92)90060-WSearch in Google Scholar

[48] Y. Vassilevski, M. Olshanskii, S. Simakov, A. Kolobov and A. Danilov, Personalized Computational Hemodynamics: Models, Methods, and Applications for Vascular Surgery and Antitumor Therapy, Academic Press, London, 2020. Search in Google Scholar

[49] H. von Wahl and T. Richter, An Eulerian time-stepping scheme for a coupled parabolic moving domain problem using equal order unfitted finite elements, Proc. Appl. Math. Mech. 22 (2023), Paper No. e202200003. 10.1002/pamm.202200003Search in Google Scholar

[50] H. von Wahl and T. Richter, Error analysis for a parabolic PDE model problem on a coupled moving domain in a fully Eulerian framework, SIAM J. Numer. Anal. 61 (2023), no. 1, 286–314. 10.1137/21M1458417Search in Google Scholar

[51] H. von Wahl, T. Richter and C. 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. 10.1093/imanum/drab044Search in Google Scholar