Splines, biharmonic operator and approximate eigenvalue

The biharmonic operator plays a central role in a wide array of physical models, such as elasticity theory and the streamfunction formulation of the Navier–Stokes equations. Its spectral theory has been extensively studied. In particular the one-dimensional case (over an interval) serves as the basic model of a high order Sturm-Liouville problem. The need for corresponding numerical simulations has led to numerous works. This review focuses on a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator (DBO). The DBO is constructed in terms of the discrete Hermitian derivative. The surprising strong connection between cubic spline functions (on an interval) and the DBO is recalled. In particular the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $\Bigl[\Bigl(\frac{d}{dx}\Bigr)^4\Bigr]^{-1}.$ This fact entails the conclusion that the eigenvalues of the DBO converge (at an “optimal” $O(h^4)$ rate) to the continuous ones. Another consequence is the validity of a comparison principle. It is well known that there is no maximum principle for the fourth-order equation. However, a positivity result is recalled, both for the continuous and the discrete biharmonic equation, claiming that in both cases the kernels are order preserving.