Spline functions, the discrete biharmonic operator and approximate eigenvalues

Abstract: The biharmonic operator plays a central role in a wide array of physical models, notably in elasticity theory and the streamfunction formulation of the Navier-Stokes equations. The need for corresponding numerical simulations has led, in recent years, to the development of a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator (DBO). The numerical results have been remarkably accurate, and have been corroborated by some rigorous proofs. However, there remained the "mystery" of the "underlying reason" for this success. This paper is a contribution in this direction, expounding the strong connection between cubic spline functions (on an interval) and the DBO. It is shown in particular that the (scaled) fourth-order distributional derivative of the cubic spline is identical to the action of the DBO on grid functions. The DBO is constructed in terms of the discrete Hermitian derivative. A remarkable fact is that the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $\Big[\Big(\frac{d}{dx}\Big)^4\Big]^{-1}.$ Explicit expressions are presented for both kernels. 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, and the discrete eigenvalues are proved to converge (at an "optimal" $O(h^4)$ rate) to the continuous ones. Another remarkable 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.