Two‐Grid Solution of Symm's Integral Equation

Abstract

We propose two-grid iteration methods for Symm's integral equation discretized by quadrature-collocation or quadrature methods. Asymptotically the optimal order of error estimate is achieved already on the fist iteration, for some modifications on the second iteration. This enables us to introduce some solvers which are of the optimal convergence order and cheap in a practical implementation; the cost varies between 0 (N') and O(N1og N ) arithmetic operations. Numerical experiments confirm the approximation properties of the schemes. doesn't equal 1, see [13], [18]. Introduce the standard parametrization zy(t> = e -l J 2 cos 2nt , z i ( t ) = e-'/'sin 2nt of the circle ro of radius e-l/'. One can decompose equation (1.2) into Integral Equations Appl., 1 (1988), 549-579 485-495

A fast Fourier-Galerkin method for solving integral equations of second kind with weakly singular kernels

“…Developing the fast method for (1.9) becomes crucial. The Fourier-Galerkin method using the trigonometric polynomial basis among the numerical methods (cf., [1][2][3][6][7][8][9]) is a standard numerical treatment for solving (1.2). In [4] they develop a fast Fourier-Galerkin method for solving the boundary integral equation.…”

On the fast and fully discretized solution¶of integral and pseudo-differential equations¶on smooth curves

“…On the basis of fully discrete Galerkin schemes and collocation methods, fast solvers, i.e., algorithms where approximations can be computed in O N log N arithmetical operations, can also be generated by two-grid iteration schemes (see [18], [19] or Saranen and Vainikko [13], [14]). Compared with those, the CGNR method has an essentially simpler computational algorithm -only matrix-vector computations are involved.…”

Fast solvers of integral and pseudodifferential equations on closed curves

“…It seems possible to reorganize these methods so that the accuracy (8) will be obtained in O(N log N ) arithmetical operations, but we cannot go into details in this paper. In a rather special case of the Symm's integral equation this was achieved in [19]; the case is covered also by [1] with the simplest preconditioner B = A (16)). So our results cover also the (fully discretized) Galerkin method applied directly to (1), provided that a ± 0 are constant.…”

“…We shall use the formulation (19), constructing a discretization of (1) and (20) to examine the convergence properties of it.…”