A new superconvergent collocation method for eigenvalue problems

Abstract

Abstract. Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4r can be obtained. This improves upon the order 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r and 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

Convergence analysis of reproducing kernel particle method to elliptic eigenvalue problem

“…However, only limited work has been published on the theory and applications of meshfree methods for solving EVPs. For example, Kulkarni [29] proposed collocation method for EVP based on strong formulation, and Chen et al [12] introduced the HP clouds approach and higher-order stabilized conforming nodal integration for solving the Schrodinger equations. The theoretical investigation of meshfree methods under Galerkin weak form for EVPs is still lacking.…”

A spectral collocation method for eigenvalue problems of compact integral operators

“…In [13], Dellwo and Friedman proposed a new approach by solving a polynomial eigenvalue problem of a higher degree, based upon which Alam et al [1] obtained an accelerated spectral approximation for eigenelements. Kulkarni [16] introduced another method by involving a new approximation operator T n and obtained a high-order convergence rate. In addition, a multiscale method was discussed in [11].…”