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

The fixed membrane problem All + Xu = 0 in Q, u = 0 on aQ, for a bounded region Q 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, 0(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 Au + qu = J in Q, u = 0 in aQ, is also given.