Iterated discrete polynomially based Galerkin methods

Golberg and Bowman [Appl. Math. Comput. 96 (1998) 237] have studied polynomially based discrete Galerkin method for Fredholm and Singular integral equations. In this paper we consider polynomially based iterated discrete Galerkin method for solution of operator equations and for eigenvalue problem associated with an integral operator with a smooth kernel. We show that the error in the infinity norm, both for approximation of operator equation and of spectral subspace, is of the order of n(-r), where n is the degree of the polynomial approximation and r is the smoothness of the kernel. Thus the iterated discrete Galerkin solution improves upon the discrete Galerkin solution, which was shown to be of order n(-r+1) by Golberg and Bowman [Appl. Math. Comput. 96 (1998) 237]. We also give a shorter proof of the result by Golberg and Bowman which states that the error in 2-norm in discrete Galerkin method is of the order of n(-r). (C) 200

A degenerate kernel method for eigenvalue problems of compact integral operators

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by a degenerate kernel method. By interpolating the kernel of the integral operator in both the variables, we prove that the error bounds for eigenvalues and for the distance between the spectral subspaces are of the orders $h^{2r}$ and $h^r$ respectively. By iterating the eigenfunctions we show that the error bounds for eigenfunctions are of the orders $h^{2r}$.We give the numerical results

Spectral approximation using iterated discrete Galerkin method

We consider approximation of eigenelements of an integral operator with a smooth kernel by discrete Galerkin and iterated discrete Galerkin methods. We prove that by using a sufficiently accurate numerical quadrature formula, the orders of convergence in Galerkin/iterated Galerkin methods are preserved. We show that we achieve the same order of convergence in iterated discrete Galerkin method as in Nystrom method, but the size of the generalised eigenvalue problem to be solved is reduced by half

Spectral refinement using a new projection method

In this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of h(4r)(h(3r))(k), where h is the mesh of the partition and k = 0, 1, 2,... denotes the step of the iteration. This order of convergence is to be compared with the orders h(2r)(h(r))k in the Galerkin and projection methods and h(2r)(h(2r))(k) in the Sloan method. The error in eigenvector iterates is shown to be of the order of h(3r)(h(3r))(k) in the new method, h(r)(h(r))(k) in the Galerkin and projection methods and h(2r)(h(2r))(k) in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples