A numerical investigation of the solution of a class of fourth-order eigenvalue problems (Q2713413)

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summerized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied. NEWLINENEWLINENEWLINEThe authors extend the numerical studies of \textit{B. M. Brown, P. K. Jimack} and \textit{M. D. Mihajlović} [An efficient direct solver for a class of mixed finite element problems, Appl. Numer. Math. (to appear)] and of \textit{P. E. Bjørstad} and \textit{B. P. Tjøstheim} [Computing 63, No. 2, 57-107 (1999; Zbl 0940.65119)], which concern the behaviour of the principal eigenfunction of the biharmonic operator near the corners of a square domain, to the case of a general internal angle. A number of results obtained from the mixed finite-element approach are presented for a variety of domain. NEWLINENEWLINENEWLINEThe paper concludes with a brief discussion of obtained results along with a consideration of the issues associated with the validation of numerical simulations such as these, including mesh convergence and the use of rigorous enclosure techniques.