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A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

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Abstract

In this paper, a new finite element and finite difference (FE-FD) method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes. In the spatial discretization, the standard \(P_1\) FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite, while near the interface, the maximum principle preserving immersed interface discretization is applied. In the time discretization, a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate. Correction terms are needed when the interface crosses grid lines. The moving interface is represented by the zero level set of a Lipschitz continuous function. Numerical experiments presented in this paper confirm second order convergence.

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Notes

  1. Some quantities in this subsection such as \(\theta ^*,\) \(\xi _k\), and \(\eta _k\), etc. depend on i and j. For simplicity of the presentation, we omit the dependence.

References

  1. Adams, L., Li, Z.: The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24, 463–479 (2002). https://doi.org/10.1137/S1064827501389849

    Article  MathSciNet  Google Scholar 

  2. Bergmann, S., Albe, K., Flegel, E., Barragan-Yani, D.A., Wagner, B.: Anisotropic solid-liquid interface kinetics in silicon: an atomistically informed phase-field model. Modell. Simul. Mater. Sci. Eng. 25(6), 065015 (2017). https://doi.org/10.1088/1361-651X/aa7862

    Article  Google Scholar 

  3. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)

    Book  Google Scholar 

  5. De Zeeuw, D.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33, 1–27 (1990). https://doi.org/10.1016/0377-0427(90)90252-U

    Article  MathSciNet  Google Scholar 

  6. Dong, B.Y., Feng, X.F., Li, Z.: An FE-FD method for anisotropic elliptic interface problems. SIAM J. Sci. Comput. 42, B1041–B1066 (2020). https://doi.org/10.1137/19M1291030

    Article  MathSciNet  Google Scholar 

  7. Dong, B.Y., Feng, X.F., Li, Z.: An L∞ second order Cartesian method for 3D anisotropic elliptic interface problems. J. Comput. Math. 40, 886–913 (2022). https://doi.org/10.4208/jcm.2103-m2020-0107

    Article  MathSciNet  Google Scholar 

  8. Evans, L. C.: Partial Differential Equations. AMS (1998)

  9. He, X., Lin, T., Lin, Y., Zhang, X.: Immersed finite element methods for parabolic equations with moving interface. Numer. Methods Partial Differential Equations 29(2), 619–646 (2013). https://doi.org/10.1002/num.21722

    Article  MathSciNet  Google Scholar 

  10. Hou, T., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134, 236–252 (1997). https://doi.org/10.1006/jcph.1997.5689

    Article  MathSciNet  Google Scholar 

  11. Huang, W., Rokhlin, S.I.: Interface waves along an anisotropic imperfect interface between anisotropic solids. J. Nondestruc. Eval. 11, 185–198 (1992). https://doi.org/10.1007/BF00566409

    Article  Google Scholar 

  12. Langer, J.S.: Instabilities and patten formation in crystal growth. Rev. Modern Phys. 52, 1–28 (1980). https://doi.org/10.1103/RevModPhys.52.1

    Article  Google Scholar 

  13. Levitas, V.I., Warren, J.A.: Phase field approach with anisotropic interface energy and interface stresses: large strain formulation. J. Mech. Phy. Solids. 91, 94–125 (2016). https://doi.org/10.1016/j.jmps.2016.02.029

    Article  MathSciNet  Google Scholar 

  14. Li, Z.: Immersed interface method for moving interface problems. Numer. Algorithm 14, 269–293 (1997). https://doi.org/10.1023/A:1019173215885

    Article  MathSciNet  Google Scholar 

  15. Li, Z., Ito, K.: Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23, 1225–1242 (2001). https://doi.org/10.1137/S1064827500370160

    Article  MathSciNet  Google Scholar 

  16. Li, Z., Soni, B.: Fast and accurate numerical approaches for Stefan problems and crystal growth. Numer. Heat Transf. B: Fundam. 35, 461–484 (1999). https://doi.org/10.1080/104077999275848

    Article  Google Scholar 

  17. Lin, T., Lin, Y., Zhang, X.: A method of lines based on immersed finite elements for parabolic moving interface problems. Adv. Appl. Math. Mech. 5(4), 548–568 (2013). https://doi.org/10.1017/S2070073300001387

    Article  MathSciNet  Google Scholar 

  18. Lin, T., Lin, Y., Zhang, X.: Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump. Contemp. Math. 586, 257–265 (2013). https://doi.org/10.1090/conm/586/11633

    Article  MathSciNet  Google Scholar 

  19. McFadden, G.B., Wheeler, A.A., Braun, R.J., Coriell, S.R., Sekerka, R.F.: Phase-field models for anisotropic interfaces. Phys. Rev. E 48, 2016–2024 (1993). https://doi.org/10.1103/PhysRevE.48.2016

    Article  MathSciNet  Google Scholar 

  20. Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations. Cambridge Press, Cambridge (1995)

    Google Scholar 

  21. Sethian, J., Straint, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992). https://doi.org/10.1016/0021-9991(92)90140-T

    Article  MathSciNet  Google Scholar 

  22. Schittkowski, K.: QL-quadratic Programming, version 1.5 (1991). https://www.uni-bayreuth.de/departments/math/~kschittkowski/ql.htm

  23. Suo, Z.: Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. A 427, 331–358 (1990)

    MathSciNet  Google Scholar 

  24. Tuncel, N. G., Serbest, A. H.: Reflection and refraction by an anisotropic metamaterial slab with diagonal anisotropy. In: 2015 IEEE International Conference on Microwaves, Communications, Antennas and Electronic Systems (COMCAS), Tel Aviv, Israel, 2015, pp. 1–4. IEEE (2015)

  25. Yang, Q., Zhang, X.: Discontinuous Galerkin immersed finite element methods for parabolic interface problems. J. Comput. Appl. Math. 299, 127–139 (2016). https://doi.org/10.1016/j.cam.2015.11.020

    Article  MathSciNet  Google Scholar 

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Acknowledgements

B. Dong is partially supported by the National Natural Science Foundation of China (Grant No. 12261070) and the Ningxia Key Research and Development Project of China (Grant No. 2022BSB03048). Z. Li is partially supported by the Simons (Grant No. 633724) and by Fundación Séneca grant 21760/IV/22. J. Ruiz is partially supported by the Spanish national research project PID2019-108336GB-I00 and by Fundación Séneca grant 21728/EE/22. (Este trabajo es resultado de las estancias (21760/IV/22) y (21728/EE/22) financiadas por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad, Colaboración Internacional e Intercambio de Conocimiento “Jiménez de la Espada”. (Plan de Actuación 2022)). We also would like to thank the referees for helpful suggestions.

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Authors and Affiliations

  1. School of Civil and Hydraulic Engineering, Ning**a University, Yinchuan, 750021, Ningxia, China

    Baiying Dong

  2. School of Mathematics and Computer Science, Ning**a Normal University, Guyuan, 756000, Ningxia, China

    Baiying Dong

  3. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    Zhilin Li

  4. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Juan Ruiz-Álvarez

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  1. Baiying Dong
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Correspondence to Zhilin Li.

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On the occasion of Dr. Stanley Osher’s 80th birthday.

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Dong, B., Li, Z. & Ruiz-Álvarez, J. A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces. Commun. Appl. Math. Comput. 6, 992–1012 (2024). https://doi.org/10.1007/s42967-023-00281-x

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