A discrete Galerkin method for first kind integral equations with a logarithmic kernel. (English) Zbl 0676.65140
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.
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.
Reviewer: G.Hecquet
MSC:
65R20 | Numerical methods for integral equations |
65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
35C15 | Integral representations of solutions to PDEs |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |