A New Superconvergent Projection Method for Approximate Solutions of Eigenvalue Problems

Abstract

We propose here a new method based on projections for approximate solution of eigenvalue problems associated with a compact linear operator. For an integral operator with a smooth kernel using the orthogonal projection onto the space of discontinuous piecewise polynomials of degree ≤r−1, we show that the new method exhibits the error of the order of 4r for eigenvalue approximation and of the order of 3r for spectral subspace approximation. This improves upon the order 2r for eigenvalue approximation in Galerkin/Iterated Galerkin method and the orders r and 2r for spectral subspace approximation in Galerkin and the Iterated Galerkin method, respectively. We illustrate this improvement in the order of convergence by a numerical example.

Keywords:

• Integral equations
• Projection
• Eigenvalue
• Spectral subspace
• AMS Subject Classification: 65R20