High precision solutions of two fourth order eigenvalue problems (Q1818412)

Using a highly accurate spectral Legendre-Galerkin method, the authors solve the biharmonic eigenvalue problem \(\Delta^2u=\lambda u\) and the buckling plate problem \(\Delta^2u= -\lambda\Delta u\) on the unit square under Dirichlet boundary conditions. In both cases they study for the first eigenfunction the nodal lines occurring near a corner. Five sign changes are computed, where the eigenfunction exhibits a self-similar pattern as one approaches the corner. The numerical results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows complete agreement between the numerical and the analytical results already from the first sign change.