A numerical existence proof of nodal lines for the first eigenfunction of the plate equation
We explain a numerical procedure to compute error bounds in \(H^2_0(\Omega)\) for eigenfunctions of elliptic eigenvalue problems of fourth order. Therefore, we compute a finite element approximation and an upper bound for the defect in \(H^{-2} (\Omega)\). Then, a theorem of Kato, eigenvalue inclusions and explicit embedding constants yield a pointwise error bound for the approximation. In order to control rounding errors, we use interval arithmetic. As an application, we prove the existence of nodal lines for the first eigenfunction of the clamped plate and for the buckling plate in a square.
elliptic eigenvalue problems of fourth order
scientific article; zbMATH DE number 938555