  It is well known that the usual mixed method for solving the biharmonic
eigenvalue problem by decomposing the operator into two Laplacians may generate
spurious eigenvalues on non-convex domains. To overcome this difficulty, we
adopt a recently developed mixed method, which decomposes the biharmonic
equation into three Poisson equations and still recovers the original solution.
Using this idea, we design an efficient biharmonic eigenvalue algorithm, which
contains only Poisson solvers. With this approach, eigenfunctions can be
confined in the correct space and thereby spurious modes in non-convex domains
are avoided. A priori error estimates for both eigenvalues and eigenfunctions
on quasi-uniform meshes are obtained; in particular, a convergence rate of
$\mathcal{O}({h}^{2\alpha})$ ($ 0<\alpha<\pi/\omega$, $\omega > \pi$ is the
angle of the reentrant corner) is proved for the linear finite element.
Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$
convergent rate for the quasi-uniform mesh with the regular refinement strategy
even on non-convex polygonal domains.
