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

This paper is concerned with the accurate numerical approximation of the sp ectral properties of the biharmonic operator on various domains in two dime nsions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfu nction is studied since it is known that this has an unbounded number of os cillations when approaching certain, types of corners on domain boundaries. Recent computational results of Bjorstad & Tjorstheim, using a highly accu rate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided s ufficient care is taken with the numerical method. We demonstrate that simi lar accuracy is also achieved using an unstructured finite-element solver w hich may be applied to problems on domains with arbitrary geometries. A num ber of results obtained from this mixed finite-element approach are then pr esented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of t he internal angle, and another family of (symmetric and non-convex) domains , for which the parity of the least eigenfunction is investigated. The pape r not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator.