Skip to main content
SpringerLink
Log in

Symmetric Homotopy Method for Discretized Elliptic Equations with Cubic and Quintic Nonlinearities

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

Abstract

Symmetry is analyzed in the solution set of the polynomial system resulted from the eigenfunction expansion discretization of semilinear elliptic equation with polynomial nonlinearity. Such symmetry is inherited from the symmetry of the continuous problem and is rooted in the dihedral symmetry \(D_4\) of the domain. Homotopies preserving such symmetry are designed to efficiently compute all solutions of the polynomial systems obtained from the discretizations for problems with cubic and quintic nonlinearities, respectively. The key points in homotopy construction are the special properties of the polynomial systems arising respectively from the discretizations of \(-\Delta u=u^3\) and \(-\Delta u=u^5\) in certain eigensubspaces. Such resulting polynomial systems are taken as start systems in the homotopies. Since only representative solution paths need to be followed, a lot of computational cost can be saved. Numerical results are presented to illustrate the efficiency.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Grundland, A.M., Tuszyski, J.A., Winternitz, P.: Group theory and solutions of classical field theories with polynomial nonlinearities. Found. Phys. 23, 633–665 (1993)

    Article  MathSciNet  Google Scholar 

  2. Brezis, H., Merle, F., Rivière, T.: Quantization effects for \(-\Delta u=u(1-\vert u\vert ^2)\) in \({ R}^2\). Arch. Ration. Mech. Anal. 126, 35–58 (1994)

    Article  MATH  Google Scholar 

  3. Choi, Y.S., Mckenna, P.J.: A mountain pass method for the numerical solutions of semilinear elliptic problems. Nonlinear Anal. Theory Methods Appl. 20, 417–437 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ding, Z.H., Costa, D., Chen, G.: A high-linking algorithm for sign-changing solutions of semilinear elliptitc equations. Nonlinear Anal. Theory Methods Appl. 38, 151–172 (1999)

    Article  MATH  Google Scholar 

  5. Li, Y., Zhou, J.X.: A minimax method for finding multiple critical points and its application to semilinear PDEs. SIAM J. Sci. Comput. 23, 840–865 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, C.M., Xie, Z.Q.: Search-extension method for multiple solutions of nonlinear problem. Comput. Math. Appl. 47, 327–343 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, C.M., Xie, Z.Q.: Search-Extension Method for Finding Multiple Solutions of Nonlinear Equations (in Chinese). Science Press, Beijing (2005)

    Google Scholar 

  8. Xie, Z., Chen, C., Xu, Y.: An improved search-extension method for computing multiple solutions of semilinear PDEs. IMA J. Numer. Anal. 25, 549–576 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhang, X., Zhang, J., Yu, B.: Eigenfunction expansion method for multiple solutions of semilinear elliptic equations with polynomial nonlinearity. SIAM J. Numer. Anal. 51, 2680–2699 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Neuberger, J.M., Swift, J.W.: Newton’ method and morse index for semilinear elliptic PDEs. Int. J. Bifurcation Chaos 11, 801–820 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Allgower, E.L., Bates, D.J., Sommese, A.J., Wampler, C.W.: Solution of polynomial system derived from differential equations. Computing 76, 1–10 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Allgower, E.L., Cruceanu, S.G., Tavener, S.: Application of numerical continuation to compute all solutions of semilinear elliptic equations. Adv. Geom. 9, 371–400 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hao, W., Hauenstein, J.D., Hu, B., Sommese, A.J.: A bootstrapping approach for computing multiple solutions of differential equations. J. Comp. Appl. Math. 258, 181–190 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Neuberger, J.M., Sieben, N., Swift, J.W.: Symmetry and automated branch following for a semilinear elliptic PDE on a fractal region. SIAM J. Appl. Dyn. Syst. 5, 476–507 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, T.Y., Sauer, T., York, J.A.: The random product homotopy and deficient polynomial systems. Numerische Mathematik 51, 481C–500 (1987)

    Article  MathSciNet  Google Scholar 

  16. Morgan, A.P., Sommese, A.J.: Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29, 123C–160 (1989)

    MATH  MathSciNet  Google Scholar 

  17. Lee, T.L., Li, T.Y., Tsai, C.H.: Hom4ps-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83, 109–133 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yu, B., Dong, B.: A hybrid polynomial system solving method for mixed trigonometric polynomial systems. SIAM J. Numer. Anal. 46, 1503–1518 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Verschelde, J., Cools, R.: Symmetric homotopy construction. J. Comp. Appl. Math. 50, 575–592 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Meravy, Pavol: Symmetric homotopies for solving systems of polynomial equations. Math. Slovaca 39, 277–288 (1989)

    MATH  MathSciNet  Google Scholar 

  21. Wampler, C.W., Sommese, A.J.: Numerical algebraic geometry and algebraic kinematics. Acta Numerica. 20, 469–567 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hao, W., Hauenstein, J.D., Hu, B., Sommese, A.J.: A three-dimensional steady-state tumor system. Appl. Math. Comput. 218, 2661–2669 (2011)

    MATH  MathSciNet  Google Scholar 

  23. Hao, W., Hauenstein, J.D., Hu, B., Liu, Y., Sommese, A.J., Zhang, Y.-T.: Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core, Nonlinear Anal. Ser. B Real World Appl. 13, 694–709 (2012)

    Article  MATH  Google Scholar 

  24. Hao, W., Hauenstein, J.D., Hu, B., Liu, Y., Sommese, A.J., Zhang, Y.-T.: Continuation along bifurcation branches for a tumor model with a necrotic core. J. Sci. Comp. 53, 395–413 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25, 251–276 (1999)

    Article  MATH  Google Scholar 

  26. Bates, D.J., Hauenstein, J.D., Sommese, A.J. and Wampler, C.W.: Bertini: software for numerical algebraic geometry. www3.nd.edu/~sommese/bertini

  27. Zhang, X., Yu, B., Zhang, J.: Proof of a conjecture on a discretized elliptic equation with cubic nonlinearity. Sci. China Ser. A 56, 1279–1286 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sommese, A.J., Wampler, C.W.: The Numerical Solution of Systems of Polynomials. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  29. Chen, L.: iFEM: an integrated finite element methods package in MATLAB, Technical Report, University of California at Irvine (2008)

Download references

Acknowledgments

The authors would like to thank Prof. Long Chen for the FEM package iFEM [29] in Matlab and would like to thank the anonymous referees for their valuable suggestions that led to improvement in the paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xuping Zhang

  2. School of Science, Dalian University of Technology, Panjin, 124221, Liaoning, People’s Republic of China

    Jintao Zhang & Bo Yu

  3. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Bo Yu

Authors
  1. Xuping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jintao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuping Zhang.

Additional information

Xuping Zhang: Supported in part by the Fundamental Research Funds for the Central Universities (DUT13RC(3)95, DUT16LK36) and in part by the National Natural Science Foundation of China (11401075).

Jintao Zhang: Supported in part by the Fundamental Research Funds for the Central Universities (DUT16LK35) and in part by the National Natural Science Foundation of China (11601063).

Bo Yu: Supported in part by the National Natural Science Foundation of China (11171051, 11571061).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Zhang, J. & Yu, B. Symmetric Homotopy Method for Discretized Elliptic Equations with Cubic and Quintic Nonlinearities. J Sci Comput 70, 1316–1335 (2017). https://doi.org/10.1007/s10915-016-0284-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0284-8

Keywords

Mathematics Subject Classification

Search

Navigation