On the piecewise constant collocation method for multidimensional weakly singular integral equations (Q1208137)

The author studies the numerical solution of the integral equation (1) \(u(x)=\int_ Gk(x,y)u(y)dy+f(x)\), \(x\in G\), with weakly singular kernel in a bounded domain \(G\subset\mathbb{R}^ n\) with piecewise smooth boundary. Typical examples are \(k(x,y)=a(x,y)| x-y|^{-r}\), \(0<r<n\), and \(k(x,y)=a(x,y)\log| x-y|\), where \(a(x,y)\) is a sufficiently smooth function. Convergence rates of the piecewise constant collocation method and the related cubature formula method for equation (1) have been proved in the author's earlier paper [Zh. Vychisl. Mat. Mat. Fiz. 31, No. 6, 832-849 (1991; Zbl 0736.65089)]. The present paper deals with algorithms of optimal computational complexity for solving the collocation equations with an accuracy preserving the convergence rate of the collocation method.