本篇論文利用Adini元解決週期邊界條件的Poisson特徵值問題。我們最主要是求解最小特徵值，而最小特徵值是由Rayleigh quotient公式所求得。我們也利用罰（penalty）技巧去求解週期邊界條件的Poisson特徵值問題。本篇論文的重點是探討最小特徵值會有超收斂的結果。原始的最小特徵值的最優收斂階數為６階。而我們利用高階有限元（Adini元）與（penalty）技巧求解最小特徵值會得到超收斂結果，收斂階數會提升至７階或８階。最後我們也會有數值結論去驗證它。

關鍵字；　Adini，Poisson，週期邊界條件，特徵值。

Adini’s elements are applied to Poisson’s eigenvalue problems in the unit square with periodical boundary conditions and the leading eigenvalues are obtained from the Rayleigh quotient. The penalty techniques are developed to copy with periodical boundary conditions, and superconvergence is also explored for leading eigenvalues. The optimal convergence O(h^6) are obtained for quasiuniform elements
(see [2, 21]). When the uniform rectangular elements are used, the superconvergence O(h^6+p) with p = 1 or p = 2 of leading eigenvalues is proved, where h is the maximal boundary length of Adini’s elements. Numerical experiments are carried to verify the analysis made.

Keywords. Adini’s elements, Poisson’s equation, periodical boundary conditions, eigenvalue problems.

1 Introduction 9
2 Periodical Boundary Conditions 10
3 Adini’s Elements for Eigenvalue Problems 13
4 Penalty Techniques 16
5 Error Analysis for Poisson’s Equation 19
6 Superconvergence for Eigenvalue Problems 24
7 Applications to Other Kinds of FEMs 33
7.1 Bi-quadratic Lagrange Elements . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8 Numerical Experiments 36
9 Concluding Remarks 43
A Exact Solution for Di erent Boundary Conditions 44
A.1 Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 44
A.2 Dirichlet and Periodical boundary conditions . . . . . . . . . . . . . . . . 46
A.3 Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 47
A.4 Neumann and Periodical boundary conditions . . . . . . . . . . . . . . . 50
A.5 Periodical boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 52

[1] A. Adini and R.W. Clough, Analysis of plate bending by the finite element method,
NSF Rept. G. 7337, 1961.
[2] I. Babuska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis,
vol. II, Finite Element Methods (Part I), Ciarlet, P.G. and Lions, J.L. eds.,
North-Holland, Amsterdan, pp. 641-787, 1991.
[3] F. Chatelin, Spectral Approximations of Linear Operators, Academic Press,
New York, 1983.
[4] C. Chen, Structure Theory of Superconvergence of Finite Elements, (in
Chinese), Hunan Science and Technology Press, 2001.
[5] C. Chen and Y. Huang, High Accuracy Theory of Finite Element Methods(in
Chinese), Hunan Science and Technology Press., 1995.
[6] C.S. Chien, B.W. Jeng, and Z.C. Li, A two-grid finite element method for elliptic
eigenvalue problems with periodical boundary conditions, Technical report, Department
of Applied Mathematics, National Chung-Hsing University, 2005.
[7] P.G. Ciarlet, Basic error estimates for elliptic problems in Finite Element Meth-
ods (Part 1) (Eds. P.G. Ciarlet and J.L. Lions), North-Holland, Amsterdam, 1991.
[8] H.T. Huang and Z.C. Li, Global superconvergence of Adini’s elements coupled with
the Tre tz method for singular problems, Engineering Analysis with Boundary Elements,
vol.27, pp.227-240, 2003.
[9] H.T. Huang, Z.C. Li and N.Yan, New error estimates of Adini’s elements for Pois-
son’s equation, Applied Numerical Mathematics, vol.50, pp.49-74, 2004.
[10] H.T. Huang, Z.C. Li and A. Zhou, New error estimates of biquadratic Lagrange
elements for Poisson’s equation, to appear in Applied Numerical Mathematics, 2005.
[11] R. Illner, O. Kairan and H. Large, Stationary solutions of quasi-linear Schrodinger-
Poisson systems, J. Di erential Equations, Vol.144, pp.1-16, 1998.
[12] F. Kikucki, Theory and examples of partial approximation in the finite element
method, Inter. J. Numer. Methods Engrg., Vol. 10, pp. 115–122, 1976.
[13] P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate blending
problem, Rev. Francaise Automat. Informat. Recherche Operationnelle Ser Rouge
Anal. Numer., Vol. 9, No. R-1, pp. 9–53, 1975.
[14] Z.C. Li, Optimal convergence rates for the combined methods of di erent finite ele-
ment methods, Numerical Methods for PDEs, vol. 8, pp. 203-220, 1992.
[15] Z.C. Li, Combined Methods for Elliptic Equations with Singularities, In-
terfaces and Infinities, Kluwer Academic Publishers, Dordrecht, Boston, 1998.
[16] Z.C. Li, C.S. Chien and H.T. Huang, E ective condition number for the finite dif-
ference methods for Poisson’s equation, to appear in J. Comput. Appl. Math..
[17] Z.C. Li, H.T. Huang and J. Huang, New stability analysis for the Tre tz method
coupled with high order FEM for singularity problem, Technical report, Department
of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, 2005.
[18] Q. Lin and N. Yan, The Construction and Analysis of High E cient FEM
(in Chinese), Hebei University Press, 1996.
[19] R.J. Melosh, Basis of derivation of matrices for direct sti ness method, AIAA J.,
Vol. 1, pp. 1631–1637, 1963.
[20] T. Miyoshi, Convergence of finite elements solutions represented by a non-
conforming basis, Kumamoto J. Sci, (Math), Vol. 9, pp. 11–20, 1972.
[21] G. Strang and G.J. Fix, An Analysis of Finite Element Method, Prentice-Hall,
1973.
[22] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press,
1965.
[23] Y.D. Yang, An Analysis of the Finite ElementMethod for Eigenvalue Prob-
lems(in Chinese), GuiZhou People’s Publishing Company, China, 2004.