On the convergence of the collocation method for nonlinear boundary integral equations (Q1334787)

The author considers nonlinear elliptic boundary value problems of the form (*) \(\Delta\Phi= 0\) in \(\Omega\), \({\partial u\over \partial n}\Bigl|_ \Gamma= G(\Phi|_ \Gamma)- f\) on \(\Gamma\) with a nonlinear term \(G\) defined by a Nemytskij operator. The solution \(\Phi\) is represented in the form \[ \Phi(x)= -(2\pi)^{-1} \int_ \Gamma u(y)\log| x- y| ds_ y,\quad x\in\Omega,\tag{**} \] with an appropriate boundary distribution \(u\). The jump relations for the single- layer potential (**) and the boundary condition imply a nonlinear boundary integral equation of the form \({(+)\;A(u)= f}\). By the use of a previous paper of the author [J. Aust. Math. Soc. Ser. B 34, No. 4, 419-438 (1993; Zbl 0780.65072)] the existence of a unique solution of (*) and of \((+)\), respectively, are stated under appropriate conditions. Collocation methods are used for solving numerically the nonlinear boundary integral equation \((+)\). For sufficiently small \(h\) the collocation method gives a unique solution \(u_ h\) converging to the unique solution of the nonlinear integral equation \((+)\). A numerical example is given.