Spline functions, the biharmonic operator and approximate eigenvalues

Abstract

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) constitutes the basic model of a high order Sturm–Liouville problem. The need for corresponding numerical simulations has led to numerous works. The present paper relies 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. However, the underlying reason for its accuracy remained unclear. This paper is a contribution in this direction, expounding the strong connection between cubic spline functions (on an interval) and the DBO. The first observation is that the (scaled) fourth-order distributional derivative of the cubic spline is identical to the action of the DBO on grid functions. It is shown that the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $$\left[ \left( \frac{d}{dx}\right) ^4\right] ^{-1}$$ , and explicit expressions are presented for both kernels. As an important application, the relation between the (infinite) set of eigenvalues of the fourth-order Sturm–Liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied. The discrete eigenvalues are proved to 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 derived, both for the continuous and the discrete biharmonic equation, showing that in both cases the kernels are order preserving.