The author considers the first integral equation \(\int_{s}g(Q)\log | P-Q| ds(Q)=h(P),\) \(P\in S\) with S the boundary of a simply- connected planar region D. A special Galerkin method with trigonometric polynomial approximants has been shown by other authors to converge exponentially when solving the above integral equation.
In this paper Galerkin’s method is further discretized by replacing the integrals with numerical integrals. The resulting discrete Galerkin method is shown to converge rapidly when the curve S and the data h are smooth.