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A nonsingular-kernel Dirichlet-to-Dirichlet mapping method for the exterior Stokes problem

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Abstract

This paper studies the finite element method for solving the exterior Stokes problem in two dimensions. A nonlocal boundary condition is defined using a nonsingular-kernel Dirichlet-to-Dirichlet (DtD) mapping, which maps the Dirichlet data on an interior circle to the Dirichlet data on another circular artificial boundary based on the Poisson integral formula of the Stokes problem. The truncated problem is then solved using the MINI-element method and a simple DtD iteration strategy, resulting into a sequence of unique and geometrically (h- independent) convergent solutions. The quasi-optimal error estimate is proved for the iterative solution at the end of the iteration process. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

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References

  1. Armentano, M.G., Stockdale, M.L.: Approximations by mini mixed finite element for the Stokes-Darcy coupled problem on curved domains. Int. J. Numer. Anal. Model. 18(2), 203–234 (2021)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1984). https://doi.org/10.1007/BF02576171

    Article  MathSciNet  MATH  Google Scholar 

  3. Askham, T., Rachh, M.: A boundary integral equation approach to computing eigenvalues of the Stokes operator. Adv. Comput. Math. 46(2), 20 (2020). https://doi.org/10.1007/s10444-020-09774-2

  4. Babu\(\breve{s}\)ka, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1973). https://doi.org/10.1007/BF01436561

  5. Bai, W.: The quadrilateral ‘Mini’ finite element for the Stokes problem. Comput. Methods Appl. Mech. Engrg. 143(1–2), 41–47 (1997). https://doi.org/10.1016/S0045-7825(96)01146-2

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, W.Z.: Artificial boundary conditions for two-dimensional incompressible viscous flows around an obstacle. Comput. Methods Appl. Mech. Engrg. 147(3–4), 263–273 (1997). https://doi.org/10.1016/S0045-7825(97)00001-7

    Article  MathSciNet  MATH  Google Scholar 

  7. Bao, W.Z.: Artificial boundary conditions for incompressible Navier-Stokes equations: a well-posed result. Comput. Methods Appl. Mech. Engrg. 188(1–3), 595–611 (2000). https://doi.org/10.1016/S0045-7825(99)00285-6

    Article  MathSciNet  MATH  Google Scholar 

  8. Bao, W.Z.: Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions. IMA J. Numer. Anal. 23(1), 125–148 (2003). https://doi.org/10.1093/imanum/23.1.125

    Article  MathSciNet  MATH  Google Scholar 

  9. Batchelor, G.K.: An introduction to fluid dynamics, paperback Cambridge Mathematical Library. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications, vol. 44. Springer (2013)

  11. Brenner, S.C.: The mathematical theory of finite element methods. Springer (2008)

  12. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151 (1974). https://doi.org/10.1051/m2an/197408R201291

  13. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, vol. 15. Springer Science & Business Media (2012)

  14. Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305 (2012). https://doi.org/10.1017/S0962492912000037

    Article  MathSciNet  MATH  Google Scholar 

  15. Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106(2), 367–413 (1985). https://doi.org/10.1016/0022-247X(85)90118-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Cuvelier, C., Segal, A., Van Steenhoven, A.A.: Finite element methods and Navier-Stokes equations, vol. 22. Springer Science & Business Media (1986)

  17. Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology: volume 1 physical origins and classical methods. Springer Science & Business Media (2012)

  18. Domínguez, V., Graham, I.G., Smyshlyaev, V.P.: A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numer. Math. 106(3), 471–510 (2007). https://doi.org/10.1007/s00211-007-0071-4

    Article  MathSciNet  MATH  Google Scholar 

  19. Domínguez, V., Sayas, F.-J.: A BEM-FEM overlapping algorithm for the Stokes equation. Appl. Math. Comput. 182(1), 691–710 (2006). https://doi.org/10.1016/j.amc.2006.04.031

    Article  MathSciNet  MATH  Google Scholar 

  20. Engleder, S., Steinbach, O.: Stabilized boundary element methods for exterior Helmholtz problems. Numer. Math. 110(2), 145–160 (2008). https://doi.org/10.1007/s00211-008-0161-y

    Article  MathSciNet  MATH  Google Scholar 

  21. Feng, X.L., Lu, X.L., He, Y.N.: Difference finite element method for the 3D steady Navier-Stokes equations. SIAM J. Numer. Anal. 61(1), 167–193 (2023). https://doi.org/10.1137/21M1450872

    Article  MathSciNet  MATH  Google Scholar 

  22. Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equations, vol. 749. Springer, Berlin (1979)

    MATH  Google Scholar 

  23. Halpern, L., Schatzman, M.: Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20(2), 308–353 (1989). https://doi.org/10.1137/0520021

    Article  MathSciNet  MATH  Google Scholar 

  24. Han, H.D., Lu, J.F., Bao, W.Z.: A discrete artificial boundary condition for steady incompressible viscous flows in a no-slip channel using a fast iterative method. J. Comput. Phys. 114(2), 201–208 (1994). https://doi.org/10.1006/jcph.1994.1160

    Article  MathSciNet  MATH  Google Scholar 

  25. Han, H.D., Bao, W.Z.: An artificial boundary condition for two-dimensional incompressible viscous flows using the method of lines. Internat. J. Numer. Methods Fluids 22(6), 483–493 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Han, H.D., Wu, X.N.: Artificial boundary method. Springer, Heidelberg; Tsinghua University Press, Beijing (2013). https://doi.org/10.1007/978-3-642-35464-9

  27. He, Y.N., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations. Appl. Numer. Math. 58(10), 1503–1514 (2008). https://doi.org/10.1016/j.apnum.2007.08.005

    Article  MathSciNet  MATH  Google Scholar 

  28. Helsing, J., Karlsson, A., Rosén, A.: Comparison of integral equations for the Maxwell transmission problem with general permittivities. Adv. Comput. Math. 47(5), 76 (2021). https://doi.org/10.1007/s10444-021-09904-4

  29. Hsiao, G.C., Wendland, W.L.: Boundary integral equations, vol. 164 of Applied Mathematical Sciences. Springer-Verlag, Berlin (2008). https://doi.org/10.1007/978-3-540-68545-6

  30. Hsiao, G.C., Xu, L.: A system of boundary integral equations for the transmission problem in acoustics. Appl. Numer. Math. 61(9), 1017–1029 (2011). https://doi.org/10.1016/j.apnum.2011.05.003

    Article  MathSciNet  MATH  Google Scholar 

  31. Jerez-Hanckes, C., Pinto, J.: Spectral Galerkin method for solving Helmholtz boundary integral equations on smooth screens. IMA J. Numer. Anal. 42(4), 3571–3608 (2022). https://doi.org/10.1093/imanum/drab074

    Article  MathSciNet  MATH  Google Scholar 

  32. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017). https://doi.org/10.1137/15M1047696

    Article  MathSciNet  MATH  Google Scholar 

  33. Kim, S., Karilla, S.J.: Microhydrodynamics: principles and selected applications. Courier Corporation (2013)

  34. Kiriaki, K., Vassilios, S.: Integral equation methods in obstacle elastic scattering. Bull. Greek Math. Soc. 45, 57–69 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Koens, L., Lauga, E.: The boundary integral formulation of Stokes flows includes slender-body theory. J. Fluid Mech. 850, 12 (2018). https://doi.org/10.1017/jfm.2018.483

  36. Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Math, Appl (1969)

    MATH  Google Scholar 

  37. Li, J., He, Y.N., Xu, H.: A multi-level stabilized finite element method for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 196(29–30), 2852–2862 (2007). https://doi.org/10.1016/j.cma.2006.12.007

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, X., Rui, H.X.: A low-order divergence-free \(H(\rm div)\)-conforming finite element method for Stokes flows. IMA J. Numer. Anal. 42(4), 3711–3734 (2022). https://doi.org/10.1093/imanum/drab080

    Article  MathSciNet  MATH  Google Scholar 

  39. Lukyanov, A.V., Shikhmurzaev, Y.D., King, A.C.: A combined BIE-FE method for the Stokes equations. IMA J. Appl. Math. 73(1), 199–224 (2008). https://doi.org/10.1093/imamat/hxm046

    Article  MathSciNet  MATH  Google Scholar 

  40. Maremonti, P.: On the Stokes equations: the maximum modulus theorem. Math. Models Methods Appl. Sci. 10(7), 1047–1072 (2000). https://doi.org/10.1142/S0218202500000537

    Article  MathSciNet  MATH  Google Scholar 

  41. Márquez, A., Meddahi, S.: New implementation techniques for the exterior Stokes problem in the plane. J. Comput. Phys. 172(2), 685–703 (2001). https://doi.org/10.1006/jcph.2001.6848

    Article  MathSciNet  MATH  Google Scholar 

  42. Meddahi, S., Sayas, F.-J.: A fully discrete BEM-FEM for the exterior Stokes problem in the plane. SIAM J. Numer. Anal. 37(6), 2082–2102 (2000). https://doi.org/10.1137/S0036142998341027

    Article  MathSciNet  MATH  Google Scholar 

  43. Nikitin, N.: Finite-difference method for incompressible Navier-Stokes equations in arbitrary orthogonal curvilinear coordinates. J. Comput. Phys. 217(2), 759–781 (2006). https://doi.org/10.1016/j.jcp.2006.01.036

    Article  MathSciNet  MATH  Google Scholar 

  44. Peng, Z.: A novel multitrace boundary integral equation formulation for electromagnetic cavity scattering problems. IEEE. T. Antenn. Propag. 63(10), 4446–4457 (2015). https://doi.org/10.1109/TAP.2015.2458328

    Article  MathSciNet  MATH  Google Scholar 

  45. Pozrikidis, C.: Boundary integral and singularity methods for linearized viscous flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992). https://doi.org/10.1017/CBO9780511624124

  46. Pozrikidis, C.: Introduction to theoretical and computational fluid dynamics. The Clarendon Press, Oxford University Press, New York (1997)

    Book  MATH  Google Scholar 

  47. Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Compt. 38(158), 437–445 (1982). https://doi.org/10.2307/2007280

    Article  MathSciNet  MATH  Google Scholar 

  48. Shao, M.R., Song, L.N., Li, P.W.: A generalized finite difference method for solving Stokes interface problems. Eng. Anal. Bound. Elem. 132, 50–64 (2021). https://doi.org/10.1016/j.enganabound.2021.07.002

    Article  MathSciNet  MATH  Google Scholar 

  49. Sun, T., Wang, J.L., Zheng, C.X.: Fast evaluation of artificial boundary conditions for advection diffusion equations. SIAM J. Numer. Anal. 58(6), 3530–3557 (2020). https://doi.org/10.1137/19M130145X

    Article  MathSciNet  MATH  Google Scholar 

  50. Sun, T., Zheng, C.X.: Efficient implementation of the exact artificial boundary condition for the exterior problem of the Stokes system in three dimensions. IMA J. Numer. Anal. 43(2), 1061–1088 (2023). https://doi.org/10.1093/imanum/drab106

    Article  MathSciNet  MATH  Google Scholar 

  51. Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55(3), 309–325 (1989). https://doi.org/10.1007/BF01390056

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, Y.S., Shi, F., You, Z.M., Zheng, H.B.: Coupled and decoupled stabilized finite element methods for the Stokes-Darcy-transport problem. J. Sci. Comput. 100(1), 7, 38 (2024). https://doi.org/10.1007/s10915-024-02534-0

  53. Wendland, W.L., Zhu, J.: The boundary element method for three-dimensional Stokes flows exterior to an open surface. Math. Comput. Model. 15(6), 19–41 (1991). https://doi.org/10.1016/0895-7177(91)90021-X

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu, L., Zhang, S., Hsiao, G.C.: Nonsingular kernel boundary integral and finite element coupling method. Appl. Numer. Math. 137, 80–90 (2019). https://doi.org/10.1016/j.apnum.2018.11.012

    Article  MathSciNet  MATH  Google Scholar 

  55. Yan, C., Cai, W., Zeng, X.: A highly scalable boundary integral equation and walk-on-spheres (BIE-WOS) method for the Laplace equation with Dirichlet data. Commun. Comput. Phys. 29(5), 1446–1468 (2021). https://doi.org/10.4208/cicp.oa-2020-0099

    Article  MathSciNet  MATH  Google Scholar 

  56. Yin, T., Hsiao, G.C., Xu, L.: Boundary integral equation methods for the two-dimensional fluid-solid interaction problem. SIAM J. Numer. Anal. 55(5), 2361–2393 (2017). https://doi.org/10.1137/16M1075673

    Article  MathSciNet  MATH  Google Scholar 

  57. Yin, T., Rathsfeld, A., Xu, L.: A BIE-based DtN-FEM for fluid-solid interaction problems. J. Comput. Math. 36(1), 47–69 (2018). https://doi.org/10.4208/jcm.1610-m2015-0480

    Article  MathSciNet  MATH  Google Scholar 

  58. Yu, D.H.: Canonical integral equations of stokes problem. J. Comput. Math. 62–73 (1986)

  59. Zhu, J., Yuan, Z.: Boundary element analysis. Science Press, Beijing (2009)

    MATH  Google Scholar 

  60. Zieniuk, E., Szerszeń, K.: NURBS curves in direct definition of the shapes of the boundary for 2D Stokes flow problems in modified classical BIE. Appl. Numer. Math. 132, 111–126 (2018). https://doi.org/10.1016/j.apnum.2018.05.013

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is partially supported by the Strategic Priority Research Program of the Chinese Academy of Sciences through Grant No. XDB0640000 and NSFC through Grants No. 12271082, 62231016, 12171465 and 12288201.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

    Xiaojuan Liu & Maojun Li

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P.R. China

    Tao Yin

  3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Xiaojuan Liu
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Communicated by: Aihui Zhou

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Liu, X., Li, M., Yin, T. et al. A nonsingular-kernel Dirichlet-to-Dirichlet mapping method for the exterior Stokes problem. Adv Comput Math 51, 2 (2025). https://doi.org/10.1007/s10444-024-10216-6

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