On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

Summary.  We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with 𝒪(n ²[log n]²) resp. 𝒪(n ²) operations, where n is the dimension of the polynomial trial space. Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002 RID="⋆" ID="⋆" Correspondence to: A. Rathsfeld Mathematics Subject Classification (1991): 65R20