Bounds for the N lowest eigenvalues of fourth-order boundary value problems

Computing (Vienna/New York)

Volumn 59, Issue 1, 1997, Pages 29-41

  Wieners, C ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Biharmonic operator; Boundary homotopy; Double eigenvalues; Eigenvalue enclosures; Plate equation

Indexed keywords

BOUNDARY VALUE PROBLEMS; ERROR ANALYSIS; ERROR CORRECTION; MATHEMATICAL OPERATORS; SPECTRUM ANALYSIS;

BIHARMONIC OPERATOR; BOUNDARY HOMOTOPY METHOD; FOURTH ORDER BOUNDARY VALUE PROBLEMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0030686292     PISSN: 0010485X     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02684402     Document Type: Article

References (20)

1. Behnke, H., Goerisch, F.: Inclusions for eigenvalues of selfadjoint problems. In: Topics in validated computations (Herzberger, J., ed.). Amsterdam: North-Holland, 1994.
2. Chatelin, F.: Spectral approximation of linear operators. New York: Academic Press, 1983.
3. Chen, G., Coleman, M.P., Zhou, J.: Analysis of vibration eigenfrequencies of a thin plate by the Keller-Rubinow wave method. I: Clamped boundary conditions with rectangular or circular geometry. SIAM J. Appl. Math. 51, 967-983 (1991).
4. Dean, E. J., Glowinski, R., Pironneau, O.: Iterative solution of the stream function - Vorticity formulation of the Stokes problem. Comp. Meth. Appl. Mech. Eng. 87, 117-155 (1991).
5. Fichera, G.: Numerical and quantitative analysis. London: Pitman 1978.
6. Frantzen, H.: Einschließungen von Eigenwerten von Platten. Diplomarbeit, Clausthal 1982.
7. Georisch, F.: Ein Stufenverfahren zur Berechnung von Eigenwertschranken, In: Numerische Behandlung von Eigenwertaufgaben (Albrecht, J., Collatz, L., Velte, W., eds,). ISNM 83, 104-117 (1984).
8. Goerisch, F., Haunhorst H.: Eigenwertschranken für Eigenwertaufgaben mit partiellen Differentialgleichungen. Z. Angew. Math. Mech. 65, 129-135 (1985).
9. Hackbusch, W., Hofmann, G.: Results of the eigenvalue problem for the plate equation. ZAMP 31, 730-739 (1980).
10. Kato, T.: On some approximate methods concerning the Operators T* T. Math. Ann. 126, 253-262 (1953).
11. Kuttler, J. R., Sigillito, V. G.: Estimating eigenvalues with a posterion/a priori inequalities. London: Pitman 1985.
12. Marcowitz, Th., Meis, U.: Numerical solution of partial differential equations. Berlin, Heidelberg, New York: Springer, 1981.
13. Plum, M.: Bounds for eigenvalues of second-order elliptic differential operators. J. Appl. Math. Phys. (ZAMP) 42, 848-863 (1991).
14. 2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165, 36-61 (1992).
15. Schwarz, H. R.: Methode der finiten Elemente. Stuttgart, 1991.
16. Wagner, B.: Eigenwertschranken für Eigenwertaufgaben der Form T*Tφ=λφ. Dissertation, Clausthal, 1992.
17. Wieners, C.: Numerische Existenzbeweise für schwache Lösungen nichtlinearer elliptischer Randwertaufgaben. Dissertation, Köln, 1994.
18. Wieners, C.: A numerical existence proof of nodal lines for the first eigenfunction of the plate equation. Arch. Math. 66, 420-427 (1996).
19. Wieners, C.: Numerical enclosures for solutions of the Navier-Stokes equation for small Reynolds numbers. In: Numerical methods and error bound (Alefeld, G., Herzberger, J., eds.), pp. 280-286. Berlin: Akademie Verlag 1996.
20. Zimmermann, S.: Uber die Genauigkeit von Eigenwertschranken für selbstadjungierte Operatoren. Dissertation, Clausthal, 1989.