The Galerkin method for singular integral equations revisited

AbstractIn an earlier work, the author has obtained error bounds for the Galerkin method for solving Cauchy singular integral equations, discovering that the usually neglected constants contain the Riemann zeta function, when evaluated in the supremum norm. The aim of this investigation is twofold: to show that the occurrence of the Riemann zeta function in the error bound for the Chebyshev norm is sharp; and secondly to use this result to obtain a class of forcing functions for which the method does not yield an approximate solution differing from the analytical one by at most a prescribed error tolerance. These counterexamples indicate that in practical situations, for functions exhibiting a behavior similar to the one presented here, Galerkin's method might not lead to an acceptable solution

Unsteady two dimensional airloads acting on oscillating thin airfoils in subsonic ventilated wind tunnels

The numerical calculation of unsteady two dimensional airloads which act upon thin airfoils in subsonic ventilated wind tunnels was studied. Neglecting certain quadrature errors, Bland's collocation method is rigorously proved to converge to the mathematically exact solution of Bland's integral equation, and a three way equivalence was established between collocation, Galerkin's method and least squares whenever the collocation points are chosen to be the nodes of the quadrature rule used for Galerkin's method. A computer program displayed convergence with respect to the number of pressure basis functions employed, and agreement with known special cases was demonstrated. Results are obtained for the combined effects of wind tunnel wall ventilation and wind tunnel depth to airfoil chord ratio, and for acoustic resonance between the airfoil and wind tunnel walls. A boundary condition is proposed for permeable walls through which mass flow rate is proportional to pressure jump

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

The numerical solution of Cauchy singular integral equations with additional fixed singularities

In this paper we propose a quadrature method for the numerical solution of Cauchy singular integral equations with additional fixed singularities. The unknown function is approximated by a weighted polynomial which is the solution of a finite dimensional equation obtained discretizing the involved integral operators by means of a Gauss-Jacobi quadrature rule. Stability and convergence results for the proposed procedure are proved. Moreover, we prove that the linear systems one has to solve, in order to determine the unknown coefficients of the approximate solutions, are well conditioned. The efficiency of the proposed method is shown through some numerical examples

Mixed boundary value problems in mechanics

Certain boundary value problems were studied over a domain D which may contain the point at infinity and may be multiply connected. Contours forming the boundary are assumed to consist of piecewise smooth arcs. Mixed boundary value problems are those with points of flux singularity on the boundary; these are points on the surface, either side of which at least one of the differential operator has different behavior. The physical system was considered to be described by two quantities, the potential and the flux type quantities. Some of the examples that were illustrated included problems in potential theory and elasticity

Spectral collocation method for compact integral operators

We propose and analyze a spectral collocation method for integral equations with compact kernels, e.g. piecewise smooth kernels and weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \; 0\u3c\mu\u3c1.$ We prove that 1) for integral equations, the convergence rate depends on the smoothness of true solutions $y(t)$. If $y(t)$ satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$ satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k$, we obtain supergeometric rate of convergence for both Volterra equations and Fredholm equations and related integro differential equations; 2) for eigenvalue problems, the convergence rate depends on the smoothness of eigenfunctions. The same convergence rate for the largest modulus eigenvalue approximation can be obtained. Moreover, the convergence rate doubles for positive compact operators. Our numerical experiments confirm our theoretical results