Guaranteed lower eigenvalue bounds for the biharmonic equation

The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconformingMorley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms. Mathematics Subject Classification (2000) 65N25 · 65N30 · 74K20 Dedicated to Dietrich Braess on the occasion of his 75th birthday. This work was supported by the DFG Research Center MATHEON. C. Carstensen (B) · D. Gallistl Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany e-mail: [email protected] D. Gallistl e-mail: [email protected] C. Carstensen Department of Computational Science and Engineering, Yonsei University, 120-749 Seoul, Korea