Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

doi:10.1007/s42967-023-00347-w

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 485-535.doi: 10.1007/s42967-023-00347-w

• ORIGINAL PAPERS • 上一篇下一篇

Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

Shipeng Mao^1,2, Jiaao Sun^1,2

1 LSEC and ICMSEC, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Beijing 100190, China;
2 Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China

收稿日期:2023-09-21 修回日期:2023-10-21 接受日期:2023-10-28 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Shipeng Mao,maosp@lsec.cc.ac.cn;Jiaao Sun,sunjiaao@lsec.cc.ac.cn E-mail:maosp@lsec.cc.ac.cn;sunjiaao@lsec.cc.ac.cn
基金资助:
This work is supported by the National Natural Science Foundation of China (Nos. 12271514, 11871467, 12161141017) and the National Key Research and Development Program of China (2023YFC3705701).

Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

Shipeng Mao^1,2, Jiaao Sun^1,2

1 LSEC and ICMSEC, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Beijing 100190, China;
2 Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2023-09-21 Revised:2023-10-21 Accepted:2023-10-28 Online:2025-06-20 Published:2025-04-21
Supported by:
This work is supported by the National Natural Science Foundation of China (Nos. 12271514, 11871467, 12161141017) and the National Key Research and Development Program of China (2023YFC3705701).

摘要/Abstract

摘要： In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the wellposedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L²-H¹ error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional L²-H¹ error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.

关键词: Shliomis model, Ferrofluids, Euler semi-implicit scheme, Mixed finite element methods, Error estimates, Unconditional convergence

Abstract: In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the wellposedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L²-H¹ error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional L²-H¹ error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.

Key words: Shliomis model, Ferrofluids, Euler semi-implicit scheme, Mixed finite element methods, Error estimates, Unconditional convergence

中图分类号:

Shipeng Mao, Jiaao Sun. Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 485-535.

参考文献

1. Abboud, H., Girault, V., Sayah, T.: A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme. Numer. Math. 114(2), 189–231 (2009)
2. Ait Ou Ammi, A., Marion, M.: Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 68(2), 189–213 (1994)
3. Amirat, Y., Hamdache, K.: Global weak solutions to a ferrofluid flow model. Math. Methods Appl. Sci. 31(2), 123–151 (2008)
4. Amirat, Y., Hamdache, K.: Strong solutions to the equations of a ferrofluid flow model. J. Math. Anal. Appl. 353(1), 271–294 (2009)
5. Amirat, Y., Hamdache, K.: Unique solvability of equations of motion for ferrofluids. Nonlinear Anal. 73(2), 471–494 (2010)
6. Amirat, Y., Hamdache, K., Murat, F.: Global weak solutions to equations of motion for magnetic fluids. J. Math. Fluid Mech. 10(3), 326–351 (2008)
7. Bao, Y., Pakhomov, A.B., Krishnan, K.M.: Brownian magnetic relaxation of water-based cobalt nanoparticle ferrofluids. J. Appl. Phys. 99(8), 08–107 (2006)
8. Behrens, S., Bönnemann, H., Modrow, H., Kempter, V., Riehemann, W., Wiedenmann, A., Odenbach, S., Will, S., Thrams, L., Hergt, R., Müller, R., Landfester, K., Schmidt, A., Schüler, D., Hempelmann, R.: Synthesis and characterization. Coll. Magn. Fluids Basics Dev. Appl. Ferrofluids 763, 1 (2009)
9. Castellanos, A.: Electrohydrodynamics. Springer, New York (1998)
10. Chen, W., Wang, S., Zhang, Y., Han, D., Wang, C., Wang, X.: Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system. IMA J. Numer. Anal. 42(3), 2621–2655 (2022)
11. Diegel, A.E., Wang, C., Wang, X., Wise, S.M.: Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system. Numer. Math. 137(3), 495–534 (2017)
12. Ding, Q., He, X., Long, X., Mao, S.: Error analysis of a fully discrete projection method for magnetohydrodynamic system. Numer. Methods Partial Differential Equations 39(2), 1449–1477 (2023)
13. Gao, H.: Optimal error estimates of a linearized backward Euler FEM for the Landau-Lifshitz equation. SIAM J. Numer. Anal. 52(5), 2574–2593 (2014)
14. Gaspari, G.D.: Bloch equation for conduction-electron spin resonance. Phys. Rev. 151, 215–219 (1966)
15. Gerbeau, J.-F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006)
16. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
17. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44/45/46/47), 6011–6045 (2006)
18. He, Y.: A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem. IMA J. Numer. Anal. 23(4), 665–691 (2003)
19. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35(2), 767–801 (2015)
20. Heister, T., Mohebujjaman, M., Rebholz, L.G.: Decoupled, unconditionally stable, higher order discretizations for MHD flow simulation. J. Sci. Comput. 71(1), 21–43 (2017)
21. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)
22. Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353–384 (1990)
23. Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20(4), 633–667 (2000)
24. Hiptmair, R., Li, L., Mao, S., Zheng, W.: A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28(04), 659–695 (2018)
25. Labovsky, A., Layton, W.J., Manica, C.C., Neda, M., Rebholz, L.G.: The stabilized extrapolated trapezoidal finite-element method for the Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198(9/10/11/12), 958–974 (2009)
26. Latorre-Esteves, M., Rinaldi, C.: Applications of magnetic nanoparticles in medicine: magnetic fluid hyperthermia. P. R. Health Sci. J. 28, 227 (2009)
27. Lavrova, O., Matthies, G., Mitkova, T., Polevikov, V., Tobiska, L.: Numerical treatment of free surface problems in ferrohydrodynamics. J. Phys.: Condens. Matter 18, 2657–2669 (2006)
28. Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10(3), 622–633 (2013)
29. Mao, S., Sun, J., Xue, W.: Unconditional convergence and error estimates of a fully discrete finite element method for the micropolar Navier-Stokes equations. J. Comput. Math. 42(1),71–110 (2024)
30. Miwa, M., Harita, H., Nishigami, T., Kaneko, R., Unozawa, H.: Frequency characteristics of stiffness and damping effect of a ferrofluid bearing. Tribol. Lett. 15, 97–105 (2003)
31. Neuringer, J.L., Rosensweig, R.E.: Ferrohydrodynamics. Phys. Fluids 7, 1927–1937 (1964)
32. Nochetto, R.H., Salgado, A.J., Tomas, I.: The equations of ferrohydrodynamics: modeling and numerical methods. Math. Models Methods Appl. Sci. 26(13), 2393–2449 (2016)
33. Nochetto, R.H., Salgado, A.J., Tomas, I.: A diffuse interface model for two-phase ferrofluid flows. Comput. Methods Appl. Mech. Engrg. 309, 497–531 (2016)
34. Nochetto, R.H., Trivisa, K., Weber, F.: On the dynamics of ferrofluids: global weak solutions to the Rosensweig system and rigorous convergence to equilibrium. SIAM J. Math. Anal. 51(6), 4245– 4286 (2019)
35. Odenbach, S.: Recent progress in magnetic fluid research. J. Phys.: Condens. Matter 16, 1135–1150 (2004)
36. Pankhurst, Q., Connolly, J., Jones, S., Dobson, J.: Applications of magnetic nanoparticles in biomedicine. J. Phys. D: Appl. Phys. 36, 1 (2003)
37. Raj, K., Moskowitz, B., Casciari, R.: Advances in ferrofluid technology. J. Magn. Magn. Mater. 149, 174–180 (1995)
38. Rannacher, R.: On Chorin’s projection method for the incompressible Navier-Stokes equations. In: The Navier-Stokes Equations II—Theory and Numerical Methods (Oberwolfach, 1991), vol. 1530, pp. 167–183. Springer, Berlin (1992)
39. Rinaldi, C., Zahn, M.: Effects of spin viscosity on ferrofluid flow profiles in alternating and rotating magnetic fields. Phys. Fluids 14, 2847–2870 (2002)
40. Rosensweig, R.E.: Ferrohydrodynamics. Mineola, New York (1985)
41. Rosensweig, R.E.: Magnetic fluids. Annu. Rev. Fluid Mech. 19, 437–463 (1987)
42. Sarwar, A., Lee, R., Depireux, D., Shapiro, B.: Magnetic injection of nanoparticles into rat inner ears at a human head working distance. IEEE Trans. Magn. 49, 440–452 (2013)
43. Shen, J.: On error estimates of projection methods for Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29(1), 57–77 (1992)
44. Shliomis, M.I.: Effective viscosity of magnetic suspensions. Sov. J. Exp. Theor. Phys. 34, 1291– 1294 (1972)
45. Shliomis, M.I.: Ferrohydrodynamics: retrospective and issues. In: Odenbach, S. (ed.) Ferrofluids: Magnetically Controllable Fluids and Their Applications. pp. 85–111. Springer, Berlin (2002)
46. Sunil, C.P., Bharti, P.K.: Double-diffusive convection in a micropolar ferromagnetic fluid. Appl. Math. Comput. 189(2), 1648–1661 (2007)
47. Temam, R.: Navier-Stokes equations: theory and numerical analysis. In: Studies in Mathematics and Its Applications, vol. 2, 3rd edn. North-Holland, Amsterdam (1984)
48. Thomée, V.: Galerkin finite element methods for parabolic problems. In: Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)
49. Torrey, H.C.: Bloch equations with diffusion terms. Phys. Rev. 104, 563–565 (1956)
50. Wang, C., Wang, J., Xia, Z., Xu, L.: Optimal error estimates of a Crank-Nicolson finite element projection method for magnetohydrodynamic equations. ESAIM: M2AN 56(3), 767–789 (2022)
51. Wu, Y., Xie, X.: Mixed finite element methods for the ferrofluid model with magnetization paralleled to the magnetic field. Numer. Math. Theory Methods Appl. 16(2), 489–510 (2023)
52. Wu, Y., Xie, X.: Energy-stable mixed finite element methods for a ferrofluid flow model. Commun. Nonlinear Sci. Numer. Simul. 125, 107330 (2023)
53. Yang, J., Mao, S.: Second order fully decoupled and unconditionally energy-stable finite element algorithm for the incompressible MHD equations. Appl. Math. Lett. 121, 107467 (2021)
54. Zahn, M., Greer, D.: Ferrohydrodynamic pumping in spatially uniform sinusoidally time-varying magnetic fields. J. Magn. Magn. Mater. 149, 165–173 (1995)
55. Zhang, G.-D., He, X., Yang, X.: Decoupled, linear, and unconditionally energy stable fully discrete finite element numerical scheme for a two-phase ferrohydrodynamics model. SIAM J. Sci. Comput. 43(1), 167–193 (2021)
56. Zhang, L.-B.: A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection. Numer. Math. Theory Methods Appl. 2(1), 65–89 (2009)
57. Zhang, L.-B., Cui, T., Liu, H.: A set of symmetric quadrature rules on triangles and tetrahedra. J. Comput. Math. 27(1), 89–96 (2009)
58. Zhang, Y., Hou, Y., Shan, L.: Numerical analysis of the Crank-Nicolson extrapolation time discrete scheme for magnetohydrodynamics flows. Numer. Methods Partial Differential Equations 31(6), 2169– 2208 (2015)

[1]	Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.
[2]	Andreas Dedner, Tristan Pryer. Discontinuous Galerkin Methods for a Class of Nonvariational Problems[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 634-656.
[3]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[4]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[5]	Leilei Wei, Shuying Zhai, Xindong Zhang. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 429-444.
[6]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[7]	Yanyong Wang, Yubin Yan, Ye Hu. Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 505-523.

Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 7

编辑推荐

Metrics

本文评价