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A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements

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Abstract

In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical PDE, namely the Cahn–Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry—and mesh—characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments.

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Notes

  1. Namely, \({\partial {u(\mu )}}/{\partial {t}}\in {L^2}[0,T;(H^1(\varOmega (\mu )){)}']\).

References

  1. ngsxfem—Add-On to NGSolve for unfitted finite element discretizations. https://github.com/ngsxfem/ngsxfem

  2. RBniCS—Reduced order modelling in FEniCS. https://www.rbnicsproject.org (2015)

  3. NGSolve—High performance multiphysics finite element software. https://github.com/NGSolve/ngsolve (2018)

  4. Agosti, A., Antonietti, P.F., Ciarletta, P., Grasselli, M., Verani, M.: A Cahn–Hilliard-type equation with application to tumor growth dynamics. Math. Methods Appl. Sci. 40(18), 7598–7626 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alikakos, N., Fusco, G., Smyrnelis, P.: Elliptic Systems of Phase Transition Type. Monograph in the Series Progress in Nonlinear Differential Equations and Their Applications, vol. 91. Birkhauser, Basel (2018)

    Book  MATH  Google Scholar 

  6. Alpak, F.O., Riviere, B., Frank, F.: A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition. Comput. Geosci. 20(5), 881–908 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30(1), 139–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Antonopoulou, D.C., Farazakis, D., Karali, G.: Malliavin calculus for the stochastic Cahn–Hilliard/Allen–Cahn equation with unbounded noise diffusion. J. Differ. Equ. 265(7), 3168–3211 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Antonopoulou, D.C., Karali, G., Millet, A.: Existence and regularity of solution for a stochastic Cahn–Hilliard/Allen–Cahn equation with unbounded noise diffusion. J. Differ. Equ. 260(3), 2383–2417 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Balajewicz, M., Farhat, C.: Reduction of nonlinear embedded boundary models for problems with evolving interfaces. J. Comput. Phys. 274, 489–504 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer. Methods Eng. 102(5), 1136–1161 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K.: Model Reduction of Parametrized Systems. MS&A Series, vol. 17. Springer, Berlin (2017)

  13. Bertozzi, A.L., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn–Hilliard equation. Trans. Image Proc. 16(1), 285–291 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Blank, L., Garcke, H., Sarbu, L., Srisupattarawanit, T., Styles, V., Voigt, A.: Phase-field Approaches to Structural Topology Optimization, pp. 245–256. Springer, Basel (2012)

    MATH  Google Scholar 

  15. Bosch, J.: Fast Iterative Solvers for Cahn–Hilliard Problems. Ph.D. thesis, Otto-von-Guericke Universität, Magdeburg (2016)

  16. Burman, E.: Ghost penalty. C. R. Math. 348(21), 1217–1220 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 52(6), 2837–2862 (2011)

    MATH  Google Scholar 

  18. Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM: M2AN 48(5–8), 859–874 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    Article  MATH  Google Scholar 

  20. Chave, F., Di Pietro, D., Marche, F., Pigeonneau, F.: A hybrid high-order method for the Cahn–Hilliard problem in mixed form. SIAM J. Numer. Anal. 54(3), 1873–1898 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cherfils, L., Fakih, H., Miranville, A.: A complex version of the Cahn–Hilliard equation for grayscale image inpainting. Multiscale Model. Simul. 15(1), 575–605 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chinesta, F., Huerta, A., Rozza, G., Willcox, K.: Model Reduction Methods, Encyclopedia of Computational Mechanics, 2nd edn., pp. 1–36. Wiley, Hoboken (2017)

    Google Scholar 

  23. Chinesta, F., Ladeveze, P., Cueto, E.: A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Eng. 18(4), 395 (2011)

    Article  Google Scholar 

  24. Choksi, R., Peletier, M., Williams, J.: On the phase diagram for microphase separation of diblock copolymers: an approach via a Nonlocal Cahn–Hilliard functional. SIAM J. Appl. Math. 69(6), 1712–1738 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chrysafinos, K., Karatzas, E.N.: Error estimates for discontinuous Galerkin time-stepping schemes for robin boundary control problems constrained to parabolic PDEs. SIAM J. Numer. Anal. 52(6), 2837–2862 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chrysafinos, K., Karatzas, E.N.: Symmetric error estimates for discontinuous Galerkin time-stepping schemes for optimal control problems constrained to evolutionary Stokes equations. Comput. Optim. Appl. 60(3), 719–751 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Claus, S., Kerfriden, P.: A CutFEM method for two-phase flow problems. Comput. Methods Appl. Mech. Eng. 348, 185–206 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Colli, P., Farshbaf-Shaker, M., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53(4), 2696–2721 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. De Groot, S., Mazur, P.: Non-equilibrium Thermodynamics. (1962), Dover Edition (2013)

  30. Dumon, A., Allery, C., Ammar, A.: Proper general decomposition (PGD) for the resolution of Navier–Stokes equations. J. Comput. Phys. 230(4), 1387–1407 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Elliott, C., Larsson, S.: Error estimates with smooth and nonsmooth data for a finite element method for the Cahn–Hilliard equation. Math. Comput. 58(S33–S36), 603–630 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Elliott, C.M.: The Cahn–Hilliard Model for the Kinetics of Phase Separation, pp. 35–73. Birkhäuser, Basel (1989)

    MATH  Google Scholar 

  33. Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn–Hilliard equation. Numer. Math. 54(5), 575–590 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Elliott, C.M., Songmu, Z.: On the Cahn–Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)

    Article  MATH  Google Scholar 

  35. Embar, A., Dolbow, J., Harari, I.: Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int. J. Numer. Methods Eng. 83(7), 877–898 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Proc. 529, 39 (1998)

    Article  MathSciNet  Google Scholar 

  37. Furihata, D., Kovàcs, M., Larsson, S., Lindgren, F.: Strong convergence of a fully discrete finite element approximation of the stochastic Cahn–Hilliard equation. SIAM J. Numer. Anal. 56(2), 708–731 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Goudenège, L., Martin, D., Vial, G.: High order finite element calculations for the Cahn–Hilliard equation. J. Sci. Comput. 52(2), 294–321 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gräßle, C., Hinze, M., Scharmacher, N.: POD for optimal control of the Cahn–Hilliard system using spatially adapted snapshots. In: Radu, F.A., Kumar, K., Berre, I., Nordbotten, J.M., Pop, I.S. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2017, pp. 703–711. Springer, Berlin (2019)

    Chapter  Google Scholar 

  40. Grepl, M., Maday, Y., Nguyen, N., Patera, A.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41(3), 575–605 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Grepl, M., Patera, A.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39(1), 157–181 (2005)

    Article  MATH  Google Scholar 

  42. Gurtin, M.E., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 06(06), 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. Math. Model. Numer. Anal. 42(2), 277–302 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Haasdonk, B., Ohlberger, M., Rozza, G.: A reduced basis method for evolution schemes with parameter-dependent explicit operators. Electron. Trans. Numer. Anal. 32, 145–161 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Harari, I., Grosu, E.: A unified approach for embedded boundary conditions for fourth-order elliptic problems. Int. J. Numer. Methods Eng. 104(7), 655–675 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics, Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  47. Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system. J. Comput. Phys. 235(C), 810–827 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hintermüller, M., Keil, T., Wegner, D.: Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system with nonmatched fluid densities. SIAM J. Control Optim. 55(3), 1954–1989 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388–418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Hinze, M., Kahle, C.: A nonlinear model predictive concept for control of two-phase flows governed by the Cahn–Hilliard Navier–Stokes system. In: Hömberg, D., Tröltzsch, F. (eds.) System Modeling and Optimization, pp. 348–357. Springer, Berlin (2013)

    Chapter  MATH  Google Scholar 

  51. Israelachvili, J.N.: Intermolecular and Surface Forces. Elsevier, Amsterdam (2011)

    Google Scholar 

  52. Jeong, D., Kim, J.: Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn–Hilliard equation. Eur. Phys. J. E 38(11), 117 (2015)

    Article  Google Scholar 

  53. Junseok, K., Seunggyu, L., Yongho, C., Seok-Min, L., Darae, J.: Basic principles and practical applications of the Cahn–Hilliard equation. Math. Probl. Eng. 1, 79–141 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Kalashnikova, I., Barone, M.F.: On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment. Int. J. Numer. Methods Eng. 83(10), 1345–1375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Karali, G., Nagase, Y.: On the existence of solution for a Cahn–Hilliard/Allen–Cahn equation. Discret. Contin. Dyn. Syst. S 7, 127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Karatzas, E.N., Ballarin, F., Rozza, G.: Projection-based reduced order models for a cut finite element method in parametrized domains. Comput. Math. Appl. 79(3), 833–851 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  57. Karatzas, E.N., Nonino, M., Ballarin, F., Rozza, G.: A Reduced order cut finite element basis for stationary and evolutionary geometrically parameterized Navier–Stokes systems, Computers & Mathematics with Applications, https://doi.org/10.1016/j.camwa.2021.07.016 (2021)

  58. Karatzas, E.N., Stabile, G., Atallah, N., Scovazzi, G., Rozza, G.: A reduced order approach for the embedded shifted boundary FEM and a heat exchange system on parametrized geometries. In: Fehr J., Haasdonk, B. (eds) IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018. IUTAM Bookseries, vol 36. Springer, Cham (2020)

  59. Karatzas, E.N., Stabile, G., Nouveau, L., Scovazzi, G., Rozza, G.: A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow. Comput. Methods Appl. Mech. Eng. 347, 568–587 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  60. Karatzas, E.N., Stabile, G., Nouveau, L., Scovazzi, G., Rozza, G.: A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 370, 113–273 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  61. Katsouleas, G., Karatzas, E.N., Travlopanos, F.: Discrete Empirical Interpolation and unfitted mesh FEMs: application in PDE-constrained optimization (2021). arXiv:2010.09059(Submitted)

  62. Katsouleas, G., Karatzas, E.N., Travlopanos, F.: Cut finite element error estimates for a class of nonlinear elliptic PDEs, pp. 1–6. Loughborough University, https://doi.org/10.17028/rd.lboro.12154854.v1, extended version at arXiv:2003.06489 (2020)

  63. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  64. Lehrenfeld, C., Reusken, A.: L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems. J. Numer. Math. 27, 85–99 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  65. Li, C., Qin, R., Ming, J., Wang, Z.: A discontinuous Galerkin method for stochastic Cahn–Hilliard equations. Comput. Math. Appl. 75(6), 2100–2114 (2018). In: 2nd Annual Meeting of SIAM Central States Section, September 30–October 2, 2016

  66. Li, Y., Jeong, D., Shin, J., Kim, J.: A conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains. Comput. Math. Appl. 65(1), 102–115 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. Luby, M.: Pseudorandomness and Cryptographic Applications. Princeton University Press, Princeton (1996). (ISBN: 9780691025469)

  68. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equi-distributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)

    Article  MATH  Google Scholar 

  69. Novick-Cohen, A., Segel, L.A.: Nonlinear aspects of the Cahn–Hilliard equation. Phys. D Nonlinear Phenom. 10(3), 277–298 (1984)

    Article  MathSciNet  Google Scholar 

  70. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations, vol. 92. UNITEXT/La Matematica per il 3+2 Book Series. Springer, Berlin (2016)

  71. Regazzoni, F., Parolini, N., Verani, M.: Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers. Comput. Methods Appl. Mech. Eng. 338, 562–596 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  72. Reshma, S., Thattil, H.J.: Inpainting of binary images using the Cahn–Hilliard equation. Int. J. Comput. Sci. Eng. Technol. 4(11), 296–300 (2014)

    Google Scholar 

  73. Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53(3), 1654–1680 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  74. Rokhzadi, A.: IMEX and Semi-implicit Runge–Kutta Schemes for CFD Simulations. Ph.D. thesis, Civil Engineering Department, Faculty of Engineering, University of Ottawa (2018)

  75. Rozza, G.: Reduced basis methods for elliptic equations in subdomains with a-posteriori error bounds and adaptivity. Appl. Numer. Math. 55(4), 403–424 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  76. Rozza, G.: Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12(1), 23–35 (2009)

    Article  MathSciNet  Google Scholar 

  77. Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  78. Rozza, G., Huynh, D.B.P., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125(1), 115–152 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  79. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  80. Schöberl, J., Arnold, A., Erb, J., Melenk, J.M., Wihler, T.P.: C++11 implementation of finite elements in NGSolve. Technical Report, Institute for Analysis and Scientific Computing, Vienna University of Technology, ASC Report 30/2014 (2014)

  81. Schott, B.: Stabilized Cut Finite Element Methods for Complex Interface Coupled Flow Problems. Ph.D. thesis, Technische Universität München (TUM) (2016)

  82. Shenyang, H.: Phase-field Models of Microstructure Evolution in a System with Elastic Inhomogeneity and Defects. Ph.D. thesis, Department of Materials Science and Engineering, Pennsylvania State University (2004)

  83. Veroy, K., Prud’homme, C., Patera, A.: Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Math. 337(9), 619–624 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wells, G.N., Kuhl, E., Garikipati, K.: A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Comput. Phys. 218(2), 860–877 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  85. Welper, G.: Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme (2019). ArXiv:1902.01079v2

  86. Wodo, O., Ganapathysubramanian, B.: Computationally efficient solution to the Cahn–Hilliard equation: adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem. J. Comput. Phys. 230(15), 6037–6060 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  87. Xu, M., Guo, H., Zou, Q.: Hessian recovery based finite element methods for the Cahn–Hilliard equation. J. Comput. Phys. 386, 524–540 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  88. Zhang, X., Li, H., Liu, C.: Optimal control problem for the Cahn–Hilliard/Allen–Cahn equation with state constraint. Appl. Math. Optim. 82, 721–754 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  89. Zhao, X., Liu, C.: Optimal control for the convective Cahn–Hilliard equation in 2D case. Appl. Math. Optim. 70(1), 61–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  90. Zhao, Y., Schillinger, D., Xu, B.X.: Variational boundary conditions based on the Nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled Cahn-Hilliard equation. J. Comput. Phys. 340, 177–199 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  91. Zhou, S., Wang, M.Y.: Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct. Multidiscip. Optim. 33(2), 89 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the European Research Council Executive Agency by means of the H2020 ERC Consolidator Grant project AROMA-CFD “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” - GA 681447, (PI: Prof. G. Rozza), FARE-X-AROMA-CFD project by MIUR, INdAM-GNCS 2018 and 2019 and by project FSE - European Social Fund - HEaD “Higher Education and Development" SISSA operazione 1, Regione Autonoma Friuli - Venezia Giulia, the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under Grant Agreement No. [1115], the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment” Grant 3270, and National Infrastructures for Research and Technology S.A. (GRNET S.A.) in the National HPC facility - ARIS - under Project ID pa190902. The authors would like also to thank Dr Andrea Mola for useful instructions regarding the cutfem stabilization, and Dr Francesco Ballarin for fruitful discussions for the reduced order part.

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Efthymios N. Karatzas

  2. FORTH Institute of Applied and Computational Mathematics, Heraklion, Crete, Greece

    Efthymios N. Karatzas

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studies, Trieste, Italy

    Efthymios N. Karatzas & Gianluigi Rozza

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Karatzas, E.N., Rozza, G. A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements. J Sci Comput 89, 9 (2021). https://doi.org/10.1007/s10915-021-01623-8

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