Collocation Approximation to Eigenvalues of an Ordinary Differential Equation: The Principle of the Thing

It is shown that simple eigenvalues of an mth order ordinary differential equation are approximated within $\mathcal {O}(|\Delta {|^{2k}})$ by collocation at Gauss points with piecewise polynomial functions of degree $< m + k$ on a mesh $\Delta$. The same rate is achieved by certain averages in case the eigenvalue is not simple. The argument relies on an extension and simplification of Osborn’s recent results concerning the approximation of eigenvalues of compact linear maps.