# Spline functions, the biharmonic operator and approximate eigenvalues

• Matania Ben-Artzi
• Guy Katriel
Article

## 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.

## Mathematics Subject Classification

Primary 34L16 Secondary 34B24 41A15 65L10

## References

1. 1.
Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and Their Applications. Academic Press, Cambridge (1967)
2. 2.
Andrew, A.L., Paine, J.W.: Correction of finite element estimates for Sturm–Liouville eigenvalues. Numer. Math. 50, 205–215 (1986)
3. 3.
Babus̆ka, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II. Elsevier, New York (1991)Google Scholar
4. 4.
Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: Navier–Stokes Equations in Planar Domains. Imperial College Press, London (2013)
5. 5.
Ben-Artzi, M., Croisille, J.-P., Fishelov, D., Katzir, R.: Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal. 38, 1485–1522 (2018)
6. 6.
Boumenir, A.: Sampling for the fourth-order Sturm–Liouville differential operator. J. Math. Anal. Appl. 278, 542–550 (2003)
7. 7.
Caudill Jr., L.F., Perry, P.A., Schueller, A.W.: Isospectral sets for fourth-order ordinary differential operators. SIAM J. Math. Anal. 29, 935–966 (1998)
8. 8.
Chawla, M.M.: A new fourth-order finite-difference method for computing eigenvalues of fourth-order two-point boundary-value problems. IMA J. Numer. Anal. 3, 291–293 (1983)
9. 9.
Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995)
10. 10.
de Boor, C.: A Practical Guide to Splines-Revised Edition. Springer, New York (2001)
11. 11.
de Boor, C., Swartz, B.: Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing. Math. Comp. 35, 679–694 (1980)
12. 12.
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, RI (1998)
13. 13.
Everitt, W.N.: The Sturm–Liouville problem for fourth-order differential equations. The Quart. J. Math. 8, 146–160 (1957)
14. 14.
Fishelov, D., Ben-Artzi, M., Croisille, J.-P.: Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equation. J. Sci. Comput. 53, 55–79 (2012)
15. 15.
Grunau, H.-C., Robert, F.: Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 196, 865–898 (2010)
16. 16.
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)
17. 17.
Kato, T.: Variation of discrete spectra. Commun. Math. Phys. 111, 501–504 (1987)
18. 18.
Lou, Z.M., Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for biharmonic problems. Numer. Math. 80, 267–303 (1998)
19. 19.
Markus, A.S.: The eigen- and singular values of the sum and product of linear operators. Russ. Math. Surv. 19, 91–120 (1964)
20. 20.
Pipher, J., Verchota, G.: A maximum principle for biharmonic functions in Lipschitz and $$C^1$$ domains. Comment. Math. Helvetici 68, 384–414 (1993)
21. 21.
Prenter, P.M.: Splines and Variational Methods. Academic Press, Cambridge (1975)
22. 22.
Rattana, A., Böckmann, C.: Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four. J. Comput. Appl. Math. 249, 144–156 (2013)
23. 23.
Schröder, J.: On linear differential inequalities. J. Math. Anal. Appl. 22, 188–216 (1968)
24. 24.
Spence, A.: On the convergence of the Nyström method for the integral equation eigenvalue problem. Numer. Math. 25, 57–66 (1975)