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Element-Free Galerkin Analysis of Stokes Problems Using the Reproducing Kernel Gradient Smoothing Integration

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Abstract

A stabilized element-free Galerkin (EFG) method is developed and analyzed in this paper for the meshless numerical solution of Stokes problems. To accelerate the solution procedure and recover the optimal convergence impaired by Gauss integration, integration constraints of Galerkin numerical methods for Stokes problems are derived, and then the inherently consistent reproducing kernel gradient smoothing integration is incorporated into the EFG method with explicit quadrature rules in the reference space. By using Nitsche’s method to satisfy the Dirichlet boundary condition, the inf-sup stability, the existence and uniqueness, and the error estimation of the EFG solution with numerical integration are derived rigorously. Theoretical results reveal that the EFG error essentially comes from not only the meshless approximations of velocity and pressure, but also numerical integration of the Galerkin weak forms. It turns out a procedure on how to choose quadrature rules to ensure that the optimal convergence is not affected by the integration error. Numerical results demonstrate the consistency, efficiency and optimal convergence of the method, and support the theoretical results.

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References

  1. Abbaszadeh, M., Dehghan, M.: Investigation of the Oldroyd model as a generalized incompressible Navier–Stokes equation via the interpolating stabilized element free Galerkin technique. Appl. Numer. Math. 150, 274–294 (2020)

    MathSciNet  MATH  Google Scholar 

  2. Álvarez Hostos, J.C., Cruchaga, M.A., Fachinotti, V.D., Carrilloa, J.A.Z., Zamora, E.: A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 372, 113380 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 12, 1–125 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Banerjee, U., Osborn, J.E., Li, Q.L.: Quadrature for meshless methods. Int. J. Numer. Methods Eng. 76, 1434–1470 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Beissel, S., Belytschko, T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139, 49–64 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2009)

    Google Scholar 

  8. Chen, J.S., Hillman, M., Chi, S.W.: Meshfree methods: progress made after 20 years. J. Eng. Mech. ASCE 143, 04017001 (2017)

    Google Scholar 

  9. Chen, J.S., Hillman, M., Rüter, M.: An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int. J. Numer. Methods Eng. 95, 387–418 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Chen, J.S., Wu, C.T., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin meshfree methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)

    MATH  Google Scholar 

  11. Choe, H.J., Kim, D.W., Kim, Y.: Meshfree method for the non-stationary incompressible Navier–Stokes equations. Discrete Contin. Dyn. Syst. Ser. B 6, 17–39 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Choe, H.J., Kim, D.W., Kim, H.H., Kim, Y.: Meshless method for the stationary incompressible Navier–Stokes equations. Discrete Contin. Dyn. Syst. Ser. B 1, 495–526 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  14. Dehghan, M., Abbaszadeh, M.: Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach. Comput. Math. Appl. 72, 427–454 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Dehghan, M., Abbaszadeh, M.: Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion. Comput. Methods Appl. Mech. Eng. 300, 770–797 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Dehghan, M., Abbaszadeh, M.: Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation. Comput. Methods Appl. Mech. Eng. 311, 856–888 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Dehghan, M., Abbaszadeh, M.: A reduced proper orthogonal decomposition (POD) element free Galerkin (POD-EFG) method to simulate two-dimensional solute transport problems and error estimate. Appl. Numer. Math. 126, 92–112 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Dehghan, M., Abbaszadeh, M.: Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method. Appl. Numer. Math. 137, 252–273 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Dehghan, M., Narimani, N.: The element-free Galerkin method based on moving least squares and moving Kriging approximations for solving two-dimensional tumor-induced angiogenesis model. Eng. Comput. 36, 1517–1537 (2020)

    Google Scholar 

  20. Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Chichester (2003)

    Google Scholar 

  21. Duan, Q.L., Gao, X., Wang, B.B., Li, X.K., Zhang, H.W., Belytschko, T., Shao, Y.L.: Consistent element-free Galerkin method. Int. J. Numer. Methods Eng. 99, 79–101 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Fernandez-Mendez, S., Huerta, A.: Imposing essential boundary conditions in mesh-free methods. Comput. Methods Appl. Mech. Eng. 193, 1257–1275 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Fries, T.P., Belytschko, T.: Convergence and stabilization of stress-point integration in mesh-free and particle methods. Int. J. Numer. Methods Eng. 74, 1067–1087 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979)

    MATH  Google Scholar 

  25. Goha, C.M., Nielsena, P.M.F., Nash, M.P.: A stabilised mixed meshfree method for incompressible media: application to linear elasticity and stokes flow. Comput. Methods Appl. Mech. Eng. 329, 575–598 (2018)

    MathSciNet  Google Scholar 

  26. Han, W.M., Meng, X.P.: Error analysis of the reproducing kernel particle method. Comput. Methods Appl. Mech. Eng. 190, 6157–6181 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Hoang, T., Verhoosel, C.V., Auricchio, F., van Brummelen, E.H., Reali, A.: Mixed isogeometric finite cell methods for the Stokes problem. Comput. Methods Appl. Mech. Eng. 316, 400–423 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Huerta, A., Vidal, Y., Villon, P.: Pseudo-divergence-free element free Galerkin method for incompressible fluid flow. Comput. Methods Appl. Mech. Eng. 193, 1119–1136 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Kamranian, M., Tatari, M., Dehghan, M.: Analysis of the stabilized element free Galerkin approximations to the Stokes equations. Appl. Numer. Math. 150, 325–340 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Kumar, V.V.K.S., Kumar, B.V.R., Das, P.C.: Weighted extended B-spline method for the approximation of the stationary Stokes problem. J. Comput. Appl. Math. 186, 335–348 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Lancaster, P., Salkauskas, K.: Surface generated by moving least squares methods. Math. Comput. 37, 141–158 (1981)

    MathSciNet  MATH  Google Scholar 

  32. Li, X.L.: Error estimates for the moving least-square approximation and the element-free Galerkin method in \(n\)-dimensional spaces. Appl. Numer. Math. 99, 77–97 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Li, X.L.: Theoretical analysis of the reproducing kernel gradient smoothing integration technique in Galerkin meshless methods. J. Comput. Math. 41, 483–506 (2023)

    MathSciNet  Google Scholar 

  34. Li, X.L., Zhu, J.L.: A meshless Galerkin method for Stokes problems using boundary integral equations. Comput. Methods Appl. Mech. Eng. 198, 2874–2885 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Li, Y.C., Liu, C., Li, W., Chai, Y.B.: Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems. Appl. Math. Comput. 442, 127755 (2023)

    MathSciNet  MATH  Google Scholar 

  36. Liu, Y., Belytschko, T.: A new support integration scheme for the weak form in mesh-free methods. Int. J. Numer. Methods Eng. 82, 699–725 (2010)

    MATH  Google Scholar 

  37. Mirzaei, D.: Analysis of moving least squares approximation revisited. J. Comput. Appl. Math. 282, 237–250 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Najafi, M., Dehghan, M., Šarler, B., Kosec, G., Mavrič, B.: Divergence-free meshless local Petrov–Galerkin method for Stokes flow. Eng. Comput. 38, 5359–5377 (2022)

    Google Scholar 

  39. Park, S.K., Jo, G., Choe, H.J.: Existence and stability in the virtual interpolation point method for the Stokes equations. J. Comput. Phys. 307, 535–549 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Song, L.N., Li, P.W., Gu, Y., Fan, C.M.: Generalized finite difference method for solving stationary 2D and 3D Stokes equations with a mixed boundary condition. Comput. Math. Appl. 80, 1726–1743 (2020)

    MathSciNet  MATH  Google Scholar 

  41. Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63, 139–148 (1995)

    MathSciNet  MATH  Google Scholar 

  42. Wan, J.S., Li, X.L.: Analysis of a superconvergent recursive moving least squares approximation. Appl. Math. Lett. 133, 108223 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Wang, D.D., Wu, J.C.: An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Comput. Methods Appl. Mech. Eng. 298, 485–519 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Wang, D.D., Wu, J.C.: An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature. Comput. Methods Appl. Mech. Eng. 349, 628–672 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Wang, L.H., Qian, Z.H., Zhou, Y.T., Peng, Y.B.: A weighted meshfree collocation method for incompressible flows using radial basis functions. J. Comput. Phys. 401, 108964 (2020)

    MathSciNet  MATH  Google Scholar 

  46. Wang, J.R., Wu, J.C., Wang, D.D.: A quasi-consistent integration method for efficient meshfree analysis of Helmholtz problems with plane wave basis functions. Eng. Anal. Bound. Elem. 110, 42–55 (2020)

    MathSciNet  MATH  Google Scholar 

  47. Wu, J.C., Wang, D.D.: An accuracy analysis of Galerkin meshfree methods accounting for numerical integration. Comput. Methods Appl. Mech. Eng. 375, 113631 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Wu, J.C., Wang, D.D., Lin, Z., Qi, D.L.: An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture. Comput. Part. Mech. 7, 193–207 (2020)

    Google Scholar 

  49. Young, D.L., Jane, S.J., Fan, C.M., Murugesan, K., Tsai, C.C.: The method of fundamental solutions for 2D and 3D Stokes problems. J. Comput. Phys. 211, 1–8 (2006)

    MATH  Google Scholar 

  50. Zhang, L., Ouyang, J., Zhang, X.H., Zhang, W.B.: On a multi-scale element-free Galerkin method for the Stokes problem. Appl. Math. Comput. 203, 745–753 (2008)

    MathSciNet  MATH  Google Scholar 

  51. Zhang, Q.H., Banerjee, U.: Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients. Adv. Comput. Math. 37, 453–492 (2012)

    MathSciNet  MATH  Google Scholar 

  52. Zhang, T., Li, X.L.: A generalized element-free Galerkin method for Stokes problem. Comput. Math. Appl. 75, 3127–3138 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Zhang, T., Li, X.L.: A Nitsche-based element-free Galerkin method for semilinear elliptic problems. Adv. Appl. Math. Mech. Accepted (2022)

  54. Zhang, T., Li, X.L., Xu, L.W.: Error analysis of an implicit Galerkin meshfree scheme for general second-order parabolic problems. Appl. Numer. Math. 177, 58–78 (2022)

    MathSciNet  MATH  Google Scholar 

  55. Zhang, Z., Hao, S.Y., Liew, K.M., Cheng, Y.M.: The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng. Anal. Bound. Elem. 37, 1576–1584 (2013)

    MathSciNet  MATH  Google Scholar 

  56. Zheng, Z.Y., Li, X.L.: Theoretical analysis of the generalized finite difference method. Comput. Math. Appl. 120, 1–14 (2022)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11971085) and the Natural Science Foundation of Chongqing (Grant Nos. cstc2021jcyj-jqX0011, cstc2021ycjh-bgzxm0065).

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  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, People’s Republic of China

    Xiaolin Li

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Li, X. Element-Free Galerkin Analysis of Stokes Problems Using the Reproducing Kernel Gradient Smoothing Integration. J Sci Comput 96, 43 (2023). https://doi.org/10.1007/s10915-023-02273-8

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