A fourth-order finite-difference approximation for the fixed membrane eigenproblem

The fixed membrane problem $\Delta u + \lambda u = 0$ in $\Omega ,u = 0$ on $\partial \Omega$, for a bounded region $\Omega$ of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type, $O({h^4})$ convergence is shown for the approximating eigenvalues and eigenfunctions, where h is the mesh width. An application to an approximation of the forced vibration problem $\Delta u + qu = f$ in $\Omega ,u = 0$ in $\partial \Omega$, is also given.