The Hermite Cubic Collocation Approximation to the EigenValues and the Eigenfuntions of the Laplace Operator

Piecewise Hermite cubics have been widely used in a variety of ways for solving partial differential equations. For a number of these techniques, knowledge about the Hermite cubic collocation approximations to the spectrum of the Laplace operator is often very useful, for error analysis and, a fortiori, possible iteration parameters. To this end, we givc here explicit closed-form expressions for the Hermite cubic approximations to both the eigenvalues and the eigenfunctions of the Laplace operator for both the Dirichlet and the Neumann problems. Moreover, for the Dirichlet case, we show that optimal approximations are obtained using the Gauss points for collocation poinL~. For both cases, we give numerical examples that verify our theoretical resull".