Skip to main content
SpringerLink
Log in

Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose and analyze spectral-Galerkin methods for the biharmonic eigenvalue problem in circular/spherical/elliptical domains. We first analyze the eigenfunction formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk/sphere to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one-dimensional problem can be obtained. Further, we extend our method to the non-separable biharmonic eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Adini, A., Clough, R.W.: Analysis of plate bending by the finite element method NSF report. G. 7337 (1961)

  2. Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72, 701–709 (1968)

    Google Scholar 

  3. Babuška, I., Osborn, J.: Eigenvalue problems, Handbook of numerical analysis. Amsterdam II, 641–787 (1991)

    MATH  Google Scholar 

  4. Belhachmi, Z., Bernardi, C., Karageorghis, A.: Spectral element discretization of the circular driven cavity, part II: the bilaplacian equation. SIAM J. Numer. Anal. 38, 1926–1960 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bialecki, B., Karageorghis, A.: A Legendre spectral Galerkin method for the biharmonic Dirichlet problem. SIAM J. on Sci. Comput. 22(5), 1549–1569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bjørstad, P.E., Tjøstheim, B.P.: Efficient algorithms for solving a fourth-order equation with spectral-Galerkin method. SIAM J. Sci. Comput. 18, 621–632 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Canuto, C.: Eigenvalue approximations by mixed methods. RAIRO-Anal. Numér. 12(1), 27–50 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, L.Z., An, J., Zhuang, Q.Q.: Direct solvers for the biharmonic eigenvalue problems using Legendre polynomials. J. Sci. Comput. (2016)

  9. Chen, W., Lin, Q.: Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method. Appl. Math. 51(1), 73–88 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davis, C.B.: A partition of unity method with penalty for fourth-order problems. J. Sci. Comput. 60, 228–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, B.Y., Shen, J., Wang, L.L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27(1-3), 305–322 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, B.Y., Wang, Z.Q., Wan, Z.S., Chu, D.L.: Second order Jacobi approximation with applications to fourth-order differential equations. Appl. Numer. Math. 55, 480–520 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, B.Y., Yu, X.H.: Spectral method for fourth-order problems on quadrilaterals. J. Sci Comput. 66(2), 477–503 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, H.: Super Spectral Viscosity Methods for Nonlinear Conservation Laws, Chebyshev Collocation Methods and Their Applications. Ph.D thesis, Shanghai University 2001; Shanghai University Press (2005)

  15. Li, H., Shen, J.: Optimal error estimates in J,acobi-weighted Sobolev spaces for polynomial approximations on the triangle. Math. Comp. 79(271), 1621–1646 (2009)

    Article  Google Scholar 

  16. Li, H., Xu, Y.: Spectral approximation on the unit ball. SIAM J. Numer Anal. 52(6), 2647–2675 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Morley, L.: The triangular equilibrium problem in the solution of plate bending problems. Aero. Quart. 19, 149–169 (1968)

    Article  Google Scholar 

  18. Osborn, J.E.: Approximation of the eigenvalues of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations. SIAM J. Numer. Anal. 13(2), 185–197 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rannacher, R.: On nonconforming and mixed finite element method for plate bending problems. The linear case. RAIRO Anal. Numer. 13, 369–387 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rappaz, J., Mercier, B., Osborn, J., Raviart, P.A.: Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36(154), 427–453 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, J.: Efficient spectral-Galerkin methods III: polar and cylindrical geometries. SIAM J. Sci. Comput. 18(6), 1583–1604 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, L., Shen, J., Wang, L.L.: Spectral approximation of time-harmonic Maxwell equations in three-dimensional exterior domains. Int. J. Numer. Anal. Model. 12(2), 1–18 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Shen, J., Tang, T., Wang, L.L.: Spectral methods: algorithms, analysis and applications. Springer Science and Business Media, Berlin (2011)

    Book  MATH  Google Scholar 

  24. Shi, Z.C.: Error estimates of Morley element. Chin. J. Numer. Math. Appl. 12, 9–15 (1990)

    MathSciNet  Google Scholar 

  25. Brenner, S.C., Monk, P., Sun, J.: C0 interior penalty Galerkin method for biharmonic eigenvalue problems. Spectral and High Order Methods for Partial Differential Equations. Lect. Notes Comput. Sci. Eng. 106, 3–15 (2015)

    Article  MATH  Google Scholar 

  26. Szegö, G.: Orthogonal polynomials. American Mathematical Society (1992)

  27. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press (1995)

  28. Yang, Y.D., Jiang, W.: Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem. Sci. China Math. 56(6), 1313–1330 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yu, X.H., Guo, B.Y.: Spectral element method for mixed inhomogeneous boundary value problems of fourth-order. J. Sci Comput. 61, 673–701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, Z.: How many numerical eigenvalues can we trust. J. Sci. Comput. 65 (2), 455–466 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is supported in part by the National Natural Science Foundation of China Grants Nos. 11661022, 11871092, and 11871455, and the Fund for Guizhou provincial colleges and universities top notch talents support program (Qianjiaohe No. KY[2018]041), the Technology Fund of Guizhou Province No. [2017]1124, and the Science and Technology Planning Project of Guizhou Province (Qiankehe Platform Talents No. [2017]5726-39).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China

    Jing An

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Jing An & Zhimin Zhang

  3. State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Huiyuan Li

  4. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Jing An
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huiyuan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhimin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing An.

Additional information

Publisher’s note

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

Appendices

Appendix A: Jacobi and generalized Jacobi polynomials

The classical Jacobi polynomials \(J_{k}^{\alpha ,\beta }(\zeta )\), k ≥ 0 with α,β > − 1 are mutually orthogonal with respect to the Jacobi weight function χα,β := χα,β(ζ) = (1 − ζ)α(1 + ζ)β on Λ = (− 1, 1):

$$ \begin{array}{@{}rcl@{}} {\int}_{-1}^{1} J_{m}^{\alpha,\beta}(\zeta) J_{n}^{\alpha,\beta}(\zeta) \chi^{\alpha,\beta}(\zeta) d\zeta = \frac{2^{\alpha+\beta+1}} {{2n+\alpha+\beta+1}}{\kern1.7pt} h^{\alpha,\beta}_{n}{\kern1.7pt} \delta_{m,n}, \quad m,n\ge 0, \end{array} $$
(A.1)

where δm,n is the Kronecker delta, and

$$ \begin{array}{@{}rcl@{}} h^{\alpha,\beta}_{n} := \frac{{\Gamma}(n+\alpha+1){\Gamma}(n+\beta+1)} {{\Gamma}(n+1){\Gamma}(n+\alpha+\beta+1)}. \end{array} $$
(A.2)

For \(k\in \mathbb {Z}\), denote by \((a)_{k}=\frac {{\Gamma }(a+k)}{{\Gamma }(a)}\) the Pochhammer symbol. The classical Jacobi polynomials possess the following important representation:

$$ \begin{array}{@{}rcl@{}} &{J}^{\alpha,\beta}_{n}(\zeta) = \sum\limits_{k=0}^{n} \frac{(-n-\beta)_{n-k} (n+\alpha+\beta+1)_{k}} {(n-k)!k!} \left( \frac{\zeta+1}{2}\right)^{k}, \end{array} $$
(A.3)

which symbolically furnishes the extension of \(J^{\alpha ,\beta }_{n}(\zeta )\) to arbitrary α and β. Generalized Jacobi polynomials preserve most of the essential properties of the classic Jacobi polynomials, among which the following identities [26] are of importance in the current paper as follows:

$$ \begin{array}{@{}rcl@{}} &&J^{\alpha,\beta}_{n}(-\zeta) =(-1)^{n} J^{\beta,\alpha}_{n}(\zeta), \end{array} $$
(A.4)
$$ \begin{array}{@{}rcl@{}} &&\partial_{\zeta} J^{\alpha,\beta}_{n}(\zeta) = \frac{n+\alpha+\beta+1}{2} J^{\alpha+1,\beta+1}_{n-1}(\zeta), \end{array} $$
(A.5)
$$ \begin{array}{@{}rcl@{}} &&J^{\alpha,\beta}_{n}(\zeta) = {\sum}_{\nu=n-k}^{n} \frac{2\nu+\alpha+\beta+k+1}{(n+\nu+\alpha+\beta+1)_{k+1}} \frac{(\nu+\beta+1)_{n-\nu}}{(\nu+\alpha+\beta+k+1)_{n-\nu-k}} \frac{(-k)_{n-\nu}}{(n-\nu)!}J^{\alpha+k,\beta}_{\nu}(\zeta), \end{array} $$
(A.6)
$$ \begin{array}{@{}rcl@{}} && (1+\zeta)J^{\alpha,\beta+1}_{n}(\zeta) = \frac{2(n+\beta+1)}{2n+\alpha+\beta+2} J^{\alpha,\beta}_{n}(\zeta) + \frac{2(n+1)}{2n+\alpha+\beta+2} J^{\alpha,\beta}_{n+1}(\zeta). \end{array} $$
(A.7)

In particular, the generalized Jacobi polynomials with α and/or β being integers are our greatest interest [15] as follows:

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} J_{n}^{\alpha,\beta}(\zeta) &=\left\{\begin{array}{ll} \left( \frac{\zeta-1}{2}\right)^{-\alpha} \left( \frac{\zeta+1}{2}\right)^{-\beta} J_{n+\alpha+\beta}^{-\alpha, -\beta}(\zeta), & \alpha,\beta\in \mathbb{Z},\ n+\alpha+\beta\in \mathbb{N}_{0},\\ h^{\alpha,\beta}_{n} \left( \frac{\zeta-1}{2}\right)^{-\alpha} J_{n+\alpha}^{-\alpha, \beta}(\zeta), & \alpha\in \mathbb{Z},\ n+\alpha\in \mathbb{N}_{0}, \\ h^{\alpha,\beta}_{n} \left( \frac{\zeta+1}{2}\right)^{-\beta} J_{n+\beta}^{\alpha, -\beta}(\zeta), & \beta\in \mathbb{Z}, \ n+\beta\in \mathbb{N}_{0}. \end{array}\right. \end{array} \end{array} $$
(A.8)

The generalized Jacobi polynomials with negative indices not only simplify the numerical analysis for the spectral approximations of differential equations but also lead to very efficient numerical algorithms [11, 23].

Finally, it is worthy to point out that a reduction of the degree of \(J^{\alpha ,\beta }_{n}(\zeta )\) occurs if and only if − n − α − β ∈{1, 2,…,n},

$$ \begin{array}{@{}rcl@{}} J^{\alpha,\beta}_{n}(\zeta) = h^{\alpha,n_{0}-n-1}_{n} J^{\alpha,\beta}_{n_{0}-1}(\zeta), \end{array} $$
(A.9)

where n0 := −n − α − β if − n − α − β ∈{1, 2,…,n} and n0 := 0 otherwise.

Appendix B: Proof of Lemma 3.2

At first, (3.13)–(3.15) are trivial results on the Jacobi expansion.

By (A.4), (A.5), and (A.8), one finds that, for i ≥ 4,

$$ \begin{array}{@{}rcl@{}} (1-t^{2})^{2} J^{2,1}_{i-4}(t) = \frac12 (1-t^{2})^{2} \left[ \frac{i}{i-2} J^{2,2}_{i-4}(t)+J^{2,2}_{i-5}(t) \right] = \frac{8 i}{i-2} J^{-2,-2}_{i}(t)+8(1-\delta_{i,4}) J^{-2,-2}_{i-1}(t). \end{array} $$

Then by (A.5), (A.4) and (A.5), one derives as follows:

$$ \begin{array}{@{}rcl@{}} &&{\partial_{t}^{2}} [(1-t^{2})^{2} J^{2,1}_{i-4}(t) ] = 2i(i-3)J^{0,0}_{i-2}(t)+2(i-4)(i-3)J^{0,0}_{i-3}(t) \\ &&\quad =\frac{2i(i-3)}{2i-3} ((i-1)J^{0,1}_{i-2}(t)+(i-2) J^{0,1}_{i-3}(t) ) +\frac{2(i-4)(i-3)}{2i-5} ((i-2)J^{0,1}_{i-3}(t)+(i-3)J^{0,1}_{i-4}(t) ) \\ &&\quad = {\frac {2 (i-3 ) (i-1 ) i}{(2i-3)}} J^{0,1}_{i-2}(t) + {\frac {8 (i-3 )^{2} (i-2 ) (i-1 )} { (2i-3 ) (2i-5 )} } J^{0,1}_{i-3}(t) + {\frac { 2(i-4 ) (i-3 )^{2}}{(2i-5)}}J^{0,1}_{i-4}(t) , \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{t+1}\partial_{t} [(1-t^{2})^{2} J^{2,1}_{i-4}(t) ] =\frac{1}{t+1}\left[ \frac{4 i (i-3)} {i-2} J^{-1,-1}_{i-1}(t)+4(i-4) J^{-1,-1}_{i-2}(t) \right] \\ &&\quad =\left[ \frac{2 i (i-3)} {i-2} J^{-1,1}_{i-2}(t)+\frac{2(i-4)(i-3)}{i-2} J^{-1,1}_{i-3}(t) \right] \\ &&\quad =\frac { 2(i-3 ) i}{(2i-3)} J^{0,1}_{i-2}(t) - \frac {12 (i-3 ) (i-2 )} { (2i-3 ) (2i-5 )} J^{0,1}_{i-3}(t) - \frac {2 (i-4 ) (i-3 )} {(2i-5)}J^{0,1}_{i-4}(t), \end{array} $$

which give (3.8) and (3.9) immediately.

Next by (A.8) and (A.5),

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{(t+1)^{2}}(1-t^{2})^{2} J^{2,1}_{i-4}(t) =(1-t)^{2} J^{2,1}_{i-4}(t) = \frac{4(i-3)}{i-1}J^{-2,1}_{i-2}(t) \\ &&\qquad = \frac {2(i-3)}{2i-3}J^{0,1}_{i-2}(t) -\frac { 8 (i-3 ) (i-2 )} { (2i-3 ) (2i-5 )} J^{0,1}_{i-3}(t) +\frac {2(i-3)}{2i-5} J^{0,1}_{i-4}(t), \end{array} $$

which states (3.11).

Further, by (A.5), (A.4), and (A.5),

$$ \begin{array}{@{}rcl@{}} &&\partial_{t}[(1-t^{2})^{2} J^{2,1}_{i-4}(t)] =\frac{4 i (i-3)} {i-2} J^{-1,-1}_{i-1}(t)+4(i-4) J^{-1,-1}_{i-2}(t) \\ &&=\frac{2(i-3)i^{2}}{(2i-3)(2i-1)} J^{0,1}_{i-1}(t) +\frac{4(i-1)(i-3)(2i^{2}-7i+2)}{(2i-5)(2i-1)(2i-3)} J^{0,1}_{i-2}(t) \\ &&-\frac{8(i-2)(i-3)}{(2i-3)(2i-5)} J^{0,1}_{i-3}(t) -\frac{4(2i^{2}-9i+6)(i-3)^{2}}{(2i-7)(2i-5)(2i-3)} J^{0,1}_{i-4}(t) -\frac{2(i-3)(i-4)^{2}}{(2i-5)(2i-7)} J^{0,1}_{i-5}(t), \end{array} $$

and by (A.8), (A.4), and (A.5),

$$ \begin{array}{@{}rcl@{}} &&(1-t)^{2}(1+t)J^{2,1}_{i-4}(t) =8J^{-2,-1}_{i-1}(t) = \frac{2i(i-3)}{(2i-1)(2i-3)} J^{0,1}_{i-1}(t) -\frac{8(i-1)(i-3)}{(2i-1)(2i-3)(2i-5)} J^{0,1}_{i-2}(t) \\ &&-\frac{4(i-2)(i-3)}{(2i-3)(2i-5)} J^{0,1}_{i-3}(t) +\frac{8(i-3)^{2}}{(2i-7)(2i-3)(2i-5)} J^{0,1}_{i-4}(t) +\frac{2(i-3))(i-4)}{(2i-5)(2i-7)} J^{0,1}_{i-5}(t), \end{array} $$

which lead to (3.10) and (3.12), respectively.

Finally, (3.16) and (3.17) are direct consequences of (3.8)–(3.15) and (A.1).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

An, J., Li, H. & Zhang, Z. Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains. Numer Algor 84, 427–455 (2020). https://doi.org/10.1007/s11075-019-00760-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00760-4

Keywords

Search

Navigation