Skip to main content
SpringerLink
Log in

Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper deals with the numerical approximation of solutions of Stokes and Brinkman systems using meshless methods. The aim is to solve a problem containing a nonzero body force, starting from the well known decomposition in terms of a particular solution and the solution of a homogeneous force problem. We propose two methods for the numerical construction of a particular solution. One method is based on the Neuber-Papkovich potentials, which we extend to nonhomogeneous Brinkman problems. A second method relies on a Helmholtz-type decomposition for the body force and enables the construction of divergence-free basis functions. Such basis functions are obtained from Hänkel functions and justified by new density results for the space H1(Ω). Several 2D numerical experiments are presented in order to discuss the feasibility and accuracy of both methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)

    MATH  Google Scholar 

  2. Alves, C.J.S., Chen, C.S.: A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23, 125–142 (2005)

    Article  MathSciNet  Google Scholar 

  3. Alves, C.J.S., Silvestre, A.L.: Density results using Stokeslets and a method of fundamental solutions for the Stokes equations. Eng. Anal. Bound. Elem. 28, 1245–1252 (2004)

    Article  Google Scholar 

  4. Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227(14), 7003–7026 (2008)

    Article  MathSciNet  Google Scholar 

  5. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22, 644–669 (1985)

    Article  MathSciNet  Google Scholar 

  6. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1949)

    Article  Google Scholar 

  7. Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions - solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118–152 (2020)

    Article  MathSciNet  Google Scholar 

  8. Evans, L.: Partial differential equations Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    Google Scholar 

  9. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

  10. Katsurada, M., Okamoto, H.: A mathematical study of the charge simulation method I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35, 507–518 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Karageorghis, A., Lesnic, D.: The method of fundamental solutions for the Oseen steady-state viscous flow past obstacles of known or unknown shapes. Numer. Methods Partial Differ. Equ. 35(6), 2103–2119 (2019)

    Article  MathSciNet  Google Scholar 

  12. Krotkiewski, M., Ligaarden, I.S., Lie, K.-A., Schmid, D.W.: On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs. Commun. Comput. Phys. 10(5), 1315–1332 (2011)

    Article  Google Scholar 

  13. Li, X.: On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Adv. Comp. Math 23, 265–277 (2005)

    Article  MathSciNet  Google Scholar 

  14. Martins, N.F.M., Rebelo, M.: Meshfree methods for nonhomogeneous Brinkman flows. Comput. Math. with Appl. 68(8), 872–886 (2014)

    Article  MathSciNet  Google Scholar 

  15. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, New-York (2000)

    MATH  Google Scholar 

  16. Nath, D., Kalra, M.S., Munshi, P.: One-stage method of fundamental and particular solutions (MFS-MPS) for the steady Navier-Stokes equations in a lid-driven cavity. Eng. Anal. Bound. Elem. 58, 39–47 (2015)

    Article  MathSciNet  Google Scholar 

  17. Neuber, H.: Ein neuer Ansatz zur lösung räumlicher Probleme der elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z. Angew. Math. Mech. 14, 203–212 (1934)

    Article  Google Scholar 

  18. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008)

    Article  MathSciNet  Google Scholar 

  19. Ogata, H., Katsurada, M.: Convergence of the invariant scheme of the method of fundamental solutions for two-dimensional potential problems in a Jordan region. Japan J. Indust. Appl. Math. 31, 231–262 (2014)

    Article  MathSciNet  Google Scholar 

  20. Papkovich, P.F.: The representation of the general integral of the fundamental equations of elasticity theory in terms of harmonic functions (in Russian). Izv. Akad. Nauk. SSSR Ser. Mat. 10, 1425–1435 (1932)

    Google Scholar 

  21. Pozrikidis, C.: Boundary integral and singularity methods for linearized viscous flows. Cambridge University Press, New-York (1992)

    Book  Google Scholar 

  22. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comp. Math 3, 251–264 (1995)

    Article  MathSciNet  Google Scholar 

  23. Sincich, E., Sarler, B.: Non-singular method of fundamental solutions based on Laplace decomposition for 2D Stokes flow problems. CMES Comput. Model. Eng. Sci. 99(5), 393–415 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Tran-Cong, T., Blake, J.R.: General solutions of the Stokes’ flow equations. J. Math. Anal. Appl. 90(1), 72–84 (1982)

    Article  MathSciNet  Google Scholar 

  25. Tsai, C.C., Young, D.L., Fan, C.M., Chen, C.W.: MFS With time-dependent fundamental solutions for unsteady Stokes equations. Eng. Anal. Bound. Elem. 30(10), 897–908 (2006)

    Article  Google Scholar 

  26. Varnhorn, W.: Boundary value problems and integral equations for the stokes resolvent in bounded and exterior domains of \(\mathbb {R}^n\), pp 206–224. World Scientific Publishing, New York (1998)

    MATH  Google Scholar 

  27. Young, D.L., Lin, Y.C., Fan, C.M., Chiu, C.L.: The method of fundamental solutions for solving incompressible Navier-Stokes problems. Eng. Anal. Bound. Elem. 33(8-9), 1031–1044 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Funding

C. J. S. Alves and A. L. Silvestre acknowledge the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID.

Author information

Authors and Affiliations

  1. Department of Mathematics and CEMAT, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Carlos J. S. Alves & Ana L. Silvestre

  2. Department of Mathematics, FCT-NOVA, Almada, Portugal

    Nuno F. M. Martins

Authors
  1. Carlos J. S. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nuno F. M. Martins
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ana L. Silvestre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nuno F. M. Martins.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by: Gianluigi Rozza

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alves, C.J.S., Martins, N.F.M. & Silvestre, A.L. Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems. Adv Comput Math 48, 44 (2022). https://doi.org/10.1007/s10444-022-09937-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-022-09937-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation