Discrete approximations of eigenvalue problems, such as finite-element, finite difference, spectral and pseudo-spectral methods, are based on the approximation of eigenfunctions by piecewise polynomial functions. These methods have a number of disadvantages such as: a) the number of eigenvalues that can be calculated is fixed and defined by the grid-step h, b) their convergence rate depends on the ordinal number of the trial eigenvalue, c) the greater the ordinal number, the worse the accuracy.
In the present paper, an alternative approach is suggested that permits overcoming the above disadvantages and obtaining approximate eigenvalues with any desired accuracy. In this so-called functional-discrete method one approximates the differential equation coefficients rather than the eigenfunctions directly. The basic idea of the method is coefficient approximation, i.e., the replacement of coefficient functions of the differential equation by a piecewise constant, piecewise linear or piecewise quadratic functions. Such approach has no restriction on the number of eigenvalues for which an approximation can be found. The convergence rate is proved. It is shown that depending on the matching point, two kinds of eigenvalue sequences may exist. For the first one, the convergence rate increases as the ordinal number of the eigenvalue increases. For the second one, the convergence rate is the same for all eigenvalues and does not depend on the ordinal number of the trial eigenvalue. The theory is checked by a number of numerical experiments.
The paper gives an extended list of references. Unfortunately, most of the referenced papers are quite old. The more recent published articles on the subject are not mentioned.