Uniform approximation of spectra of linear second order differential operators via discrete sine transform based discretisation
We study various discretisation schemes for the regular Sturm--Liouville operator on a bounded interval and for the Laplace operator on an arbitrary planar domain, in both cases subject to zero Dirichlet boundary conditions. The objective is to identify a discretisation scheme such that the corresponding discretised operator---a matrix of size $N \times N$---produces $N$ eigenvalues that approximate as closely as possible the first $N$ eigenvalues of the corresponding operator at the continuous level. By means of numerical experiments we examine several conventional discretisation schemes, and we corroborate the known fact that the conventional discretisations fail to achieve the objective, with the failure attributable to the poor approximation of high-index eigenvalues, and, as a result, to the non-uniform spectrum approximation. In contrast, the newly proposed discrete sine transform based discretisation scheme is designed in such a way that it replicates the asymptotic behaviour of high-index eigenvalues, thereby providing the sought uniform spectrum approximation. Of equal significance is the fact that the proposed discrete sine transform based scheme can be, unlike many Fourier transform based methods, applied to non-rectangular domains.