Fast solvers of integral and pseudodifferential equations on closed curves

Abstract

Abstract. On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order u N − u λ ≤ c λ,µ N λ−µ u µ, λ ≤ µ (Sobolev norms of periodic functions) in O(N log N ) arithmetical operations.

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.…”

A fast solver for a hypersingular boundary integral equation

“…In this section we will choose a quadrature rule so that the optimal convergence order of the fast Galerkin algorithm is preserved with a low complexity. To this end, we introduce the following result from [12]. By replacing g in Eq.…”

“…(1.3) is called the hypersingular boundary integral equation. The Galerkin method for the boundary integral equation using the trigonometric polynomial basis is a conventional numerical treatment (see [1][2][3]5,[7][8][9][10][11][12][13]15]). However, using this method yields a linear system with a dense coefficient matrix, that is to say, the coefficient matrix has the number of O(n 2 ) nonzero element, where 2n denotes the order of the linear system.…”

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.…”

“…estimate (14) for the approximation properties of the interpolation projector Q N . This completes the proof.…”