A polynomial collocation method for Cauchy singular integral equations over the interval
The authors consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of the integral equation is supposed to be of the form \(R:=aI + \mu^{-1}bS\mu I + K\), where \(S\) is the Cauchy integral operator, \(a\) and \(b\) are piecewise continuous coefficients, \(K\) is a regular integral operator, and \(\mu\) is a Jacobi weight. The collocation method is applied to the equation \(Ru=f\), where the collocation points are the Chebyshev nodes of the second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation in weighted \(L^2\) spaces, necessary and sufficient conditions are derived. To show stability and convergence, a general technique due to \textit{S. Roch} [Lokale Theorie des Reduktionsverfahrens für singuläre Integraloperatoren mit Carlemanschen Verschiebungen Diss. (1988; Zbl 0727.47038)] and \textit{S. Prössdorf} and \textit{B. Silbermann} [Numerical analysis for integral and related operator equations. Basel: Birkhäuser (1991; Zbl 0763.65103)] is applied. The authors introduce a special Banach algebra \(\mathcal F\) of sequences of discretized operators such that the stability of a sequence is equivalent to the invertibility of four limit operators and to the invertibility of a corresponding coset in a suitable quotient algebra \(\mathcal F/\mathcal J\). In particular, the collocation sequence \(\{A_n\}\) is shown to be an element of the algebra \(\mathcal F\). To show the invertibility of the corresponding element in the quotient algebra, a subalgebra \(\mathcal A/\mathcal J\) of this quotient algebra and a subalgebra inthe center of \(\mathcal A/\mathcal J\) are introduced. Using the Allan-Douglas local principle [cf. \textit{R. G. Douglas}, Banach algebra techniques in operator theory. New York: Academic Press (1972; Zbl 0247.47001)] the invertibility is reduced to localized problems which are then solved. It is shown that the invertibility of the four limit operators is just the stability criterion.
scientific article; zbMATH DE number 1893813