  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 summarized and their implications for numerical approximation are
discussed. In particular, the asymptotic behaviour of the first eigenfunction
is studied since it is known that this has an unbounded number of oscillations
when approaching certain types of corner on domain boundaries. Recent
computational results of Bj\o rstad and Tj\o stheim, using a highly accurate
spectral Legendre-Galerkin method, have demonstrated that a number of these
sign changes may be accurately computed on a square domain provided sufficient
care is taken with the numerical method. We demonstrate that similar accuracy
is also achieved using an unstructured finite element solver which may be
applied to problems on domains with arbitrary geometries. A number of results
obtained from this mixed finite element approach are then presented for a
variety of domains. These include a family of circular sector regions, for
which the oscillatory behaviour is studied as a function of the internal angle,
and another family of (symmetric and non-convex) domains, for which the parity
of the least eigenfunction is investigated. The paper not only verifies
existing asymptotic theory, but also allows us to make a new conjecture
concerning the eigenfunctions of the biharmonic operator.
