Skip to main content
Springer Nature Link
Log in

A High Order Compact FD Framework for Elliptic BVPs Involving Singular Sources, Interfaces, and Irregular Domains

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

High order methods are preferred in many applications such as Helmholtz equations with large wave numbers to resolve the solution numerically. In this paper, a third order compact immersed interface method (IIM) based on the standard nine-point stencil is first proposed for solving Poisson/Helmholtz interface problems with discontinuous solutions and fluxes in two-space dimensions. Theoretically, new high order jump relations are derived, which are necessary for determining the correction terms of the finite difference scheme near or on an interface. Then, based on the developed third order compact IIM, an augmented third order compact finite difference method is further developed for elliptic interface problems with piecewise constant but discontinuous coefficients. In this approach, the jump in the normal derivative is set as an unknown so that the high order compact IIM can be applied. The co-dimension one augmented variable is solved by the Schur complement system via the GMRES iterative method. Various non-trivial examples are provided to show the performance of the new methods. One important feature of the new methods is that the computed normal derivative is also nearly third order accurate. Finally, the third order augmented method is applied to Poisson/Helmholtz equations on irregular domains with few changes along examples of Neumann, Robin, and Dirichlet boundary conditions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Here we have omitted i and j dependencies for some quantities to simplify the notations if there is no confusion occurs.

  2. This is not an essential condition.

References

  1. Beale, J.T., Layton, A.T.: On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci 1, 91–119 (2006)

    Article  MathSciNet  Google Scholar 

  2. Berthelsen, P.A.: A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys. 197, 364–386 (2004)

    Article  MathSciNet  Google Scholar 

  3. De Zeeuw, D.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33, 1–27 (1990)

    Article  MathSciNet  Google Scholar 

  4. Devendran, D., Graves, D.T., Johansen, H., Ligocki, T.: A fourth-order Cartesian grid embedded boundary method for Poisson’s equation. Commun. Appl. Math. Comput. Sci. 12, 51–79 (2017)

  5. Hellrung, J., Wang, L., Sifakis, E., Teran, J.: A second-order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J. Comput. Phys. 231, 2015–2048 (2012)

    Article  MathSciNet  Google Scholar 

  6. Ito, K., Kyei, Y., Li, Z.: Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM J. Sci. Comput. 27, 346–367 (2005)

    Article  MathSciNet  Google Scholar 

  7. Le, D.V., Khoo, B.C., Peraire, J.: An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. J. Comput. Phys. 220, 109–138 (2006)

    Article  MathSciNet  Google Scholar 

  8. LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)

    Article  MathSciNet  Google Scholar 

  9. Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 230–254 (1998)

    Article  MathSciNet  Google Scholar 

  10. Li, Z., Ito, K.: Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23, 1225–1242 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Li, Z., Ito, K.: The immersed interface method—numerical solutions of PDEs involving interfaces and irregular domains. In: SIAM Frontier Series in Applied mathematics, FR33 (2006)

  12. Li, Z., Ji, H., Chen, X.: Accurate solution and gradient computation for elliptic interface problems with variable coefficients. SIAM J. Numer. Anal. 55(2), 570–597 (2016)

    Article  MathSciNet  Google Scholar 

  13. Li, Z., Qiao, Z., Tang, T.: An Introduction to Finite Difference and Finite Element Methods for ODE/PDEs of Boundary Value Problems. Cambridge University Press (2017)

  14. Liu, X., Fedkiw, R., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domain. J. Comput. Phys. 160, 151–178 (2000)

  15. Mayo, A.: A decomposition finite difference method for the fourth order accurate solution of Poisson’s equation on general reqions. Int. J. High Speed Comput. 3, 89–105 (1991)

  16. Medvinsky, M., Tsynkov, S., Turkel, E.: The method of difference potentials for the Helmholtz equation using compact high order schemes. J. Sci. Comput. 53, 150–193 (2012)

    Article  MathSciNet  Google Scholar 

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

  18. Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  MathSciNet  Google Scholar 

  19. Pan, K., Tan, Y., Hu, H.: An interpolation matched interface and boundary method for elliptic interface problems. J. Comput. Appl. Math. 234, 73–94 (2010)

    Article  MathSciNet  Google Scholar 

  20. Rutka, V., Wiegmann, A.: Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials. Numer. Algorithms 43, 309–330 (2007)

    Article  MathSciNet  Google Scholar 

  21. Tong, F., Wang, W., Zhao, J., Feng, X., Li, Z.: How to obtain an accurate gradient for interface problems? J. Comput. Phys. 405, 109070 (2020)

    Article  MathSciNet  Google Scholar 

  22. Wei, H., Luo, A., Qiu, T., Luo, R., Qi, R.: Improved Poisson-Boltzmann methods for high-performance computing. J. Chem. Theory Comput. 15, 6190–6202 (2019)

    Article  Google Scholar 

  23. Xie, Y., Ying, W.: A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions. J. Comput. Phys. 415, 109526 (2020)

    Article  MathSciNet  Google Scholar 

  24. Xu, S., Wang, Z.J.: An immersed interface method for simulating the interaction of a fluid with moving boundaries. J. Comput. Phys. 216, 454–493 (2006)

    Article  MathSciNet  Google Scholar 

  25. Ying, W.-J., Henriquez, C.S.: A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. 227, 1046–1074 (2007)

    Article  MathSciNet  Google Scholar 

  26. Zhang, Q.: Fourth- and higher-order interface tracking via mapping and adjusting regular semianalytic sets represented by cubic splines. SIAM J. Sci. Comput. 40, A3755–A3788 (2018)

    Article  MathSciNet  Google Scholar 

  27. Zhou, Y.C., Zhao, S., Feig, M., Wei, G.W.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213, 1–30 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

K. Pan is supported by Science Challenge Project (No. TZ2016002), the National Natural Science Foundation of China (No. 41874086), the Excellent Youth Foundation of Hunan Province of China (No. 2018JJ1042). D. He is supported by the president’s fund-research start-up fund from the Chinese University of Hong Kong, Shenzhen (PF01000857). Z. Li is partially supported by a Simons Grant 633724.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, 410083, China

    Kejia Pan

  2. School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, China

    Dongdong He

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

    Zhilin Li

Authors
  1. Kejia Pan
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Dongdong He
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Zhilin Li
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Zhilin Li.

Additional information

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

Pan, K., He, D. & Li, Z. A High Order Compact FD Framework for Elliptic BVPs Involving Singular Sources, Interfaces, and Irregular Domains. J Sci Comput 88, 67 (2021). https://doi.org/10.1007/s10915-021-01570-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01570-4

Keywords

Mathematics Subject Classification