Rekha P. Kulkarni

Modified projection and the iterated modified projection methods for nonlinear integral equations

Consider a nonlinear operator equation x−K(x) = f, where K is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree ≤ r, previous authors have established an order r + 1 convergence for the Galerkin solution and 2r + 2 for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order 3r + 3 and one step of iteration is shown to improve the order of convergence to 4r + 4. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

On Improvement of the Iterated Galerkin Solution of the Second Kind Integral Equations

For a second kind integral equation with a kernel which is less smooth along the diagonal, an approximate solution obtained by using a method proposed by the author in an earlier paper, is shown to have a higher rate of convergence than the iterated Galerkin solution. The projection is chosen to be either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials. The size of the system of equations that needs to be solved, in order to compute the proposed solution, remains the same as in the Galerkin method. The improvement of the proposed solution is illustrated by a numerical example.

Accelerated spectral approximation

A systematic development of higher order spectral analysis, introduced by Dellwo and Friedman, is undertaken in the framework of an appropriate product space. Accelerated analogues of Osborns results about spectral approximation are presented. Numerical examples are given by considering an integral operator.

Solution of a Schrödinger equation by iterative refinement

A simple eigenvalue and a corresponding wavefunction of a Schrodinger operator is initially approximated by the Galerkin method and by the iterated Galerkin method of Sloan. The initial approximation is iteratively refined by employing three schemes: the Rayleigh-Schrodinger scheme, the fixed point scheme and a modification of the fixed point scheme. Under suitable conditions, convergence of these schemes is established by considering error bounds. Numerical results indicate that the modified fixed point scheme along with Sloans method performs better than the others.

On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series

SummaryError bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.

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 D 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.

Asymptotic expansions for approximate solutions of Fredholm integral equations with Green's function type kernels

Asymptotic expansions at the node points for approximate solutions of second kind Fredholm integral equation with kernel of Green’s type function are obtained in the Nyström method based on composite midpoint, composite Simpson and composite modified Simpson rules. Similar expansions are obtained also for the iterated collocation method associated with piecewise constant, piecewise linear and piecewise quadratic functions. Richardson extrapolation is used to obtain approximate solutions with higher order of convergence at node/partition points. Numerical results are given to illustrate various results.

Extrapolation Using a Modified Projection Method

In [14], a new method based on projections onto a space of piecewise polynomials of degree ≤r − 1 has been shown to give a convergence of order 4r for second-kind integral equations. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin/collocation method. In this article the solution obtained by the proposed method is shown to have an asymptotic series expansion which remains valid in the discrete version. The Richardson extrapolation can then be used to further improve the order of convergence to 4r + 2.

Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced Resolvents

Block reduced resolvents are often employed in iterative schemes for refining crude approximations of the arithmetic mean of a cluster of eigenvalues and of a basis of the corresponding spectral subspace. We prove that if the bases of approximate spectral subspaces are chosen in such a way that they are bounded and each element of the basis is bounded away from the span of the previously chosen elements, then the corresponding adjoint bases are also bounded. We give an integral representation of the associated block reduced resolvent and show that under such a choice of the bases, the approximate block reduced resolvents are bounded as well. This is crucial in obtaining error estimates for the iterates of several refinement schemes. In the framework of a canonical discretization procedure for finite rank operators, appropriate choices of ises are given for various finite rank approximation methods such as Projection, Sloan, Galerkin, Nystrom, Fredholm, Degenerate kernel. If the bases are not chosen approp...

Modified projection method for Urysohn integral equations with non-smooth kernels

Consider a nonlinear operator equation x - K ( x ) = f , where K is a Urysohn integral operator with a Greens function type kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials, previous authors have investigated approximate solution of this equation using the Galerkin and the iterated Galerkin methods. They have shown that the iterated Galerkin solution is superconvergent. In this paper, a solution obtained using the iterated modified projection method is shown to converge faster than the iterated Galerkin solution. The improvement in the order of convergence is achieved by retaining the size of the system of equations same as for the Galerkin method. Numerical results are given to illustrate the improvement in the order of convergence.