Kress, Rainer; Sloan, Ian H. On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. (English) Zbl 0791.65089 Numer. Math. 66, No. 2, 199-214 (1993). A quadrature method is described for the numerical solution of the logarithmic integral equation of the first kind arising from the single- layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. An error analysis is developed in a Sobolev space setting and fast convergence rates for smooth boundary data are proven. Reviewer: R.Kress (Göttingen) Cited in 36 Documents MSC: 65N38 Boundary element methods for boundary value problems involving PDEs 65R20 Numerical methods for integral equations 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving 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) Keywords:quadrature method; logarithmic integral equation of the first kind; Dirichlet problem; Helmholtz equation; error analysis; Sobolev space; convergence rates PDFBibTeX XMLCite \textit{R. Kress} and \textit{I. H. Sloan}, Numer. Math. 66, No. 2, 199--214 (1993; Zbl 0791.65089) Full Text: DOI EuDML References: [1] Atkinson, K.E. (1988): A discrete Galerkin method for first kind integral equations with a logarithmic kernel. J. Integral Equations Appl.1, 343-363 · Zbl 0676.65140 · doi:10.1216/JIE-1988-1-3-343 [2] Atkinson, K.E., Sloan, I.H. (1991): The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math. Comput.56, 119-139 · Zbl 0713.65097 · doi:10.1090/S0025-5718-1991-1052084-0 [3] Chandler, G.A., Sloan, I.H. (1990): Spline qualocation methods for boundary integral equations. Numer. Math.58, 537-567 · Zbl 0694.65068 · doi:10.1007/BF01385639 [4] Colton, D., Kress, R. (1983): Integral Equation Methods in Scattering Theory. Wiley, New York · Zbl 0522.35001 [5] Kress, R. (1989): Linear Integral Equations. Springer Berlin Heidelberg New York · Zbl 0671.45001 [6] Kress, R. (1990): A Nyström method for boundary integral equations in domains with corners. Numer. Math.58, 145-161 · Zbl 0707.65078 · doi:10.1007/BF01385616 [7] Kress, R. (1991): Boundary integral equations in time-harmonic acoustic scattering. Math. Comput. Modelling15, 229-243 · Zbl 0731.76077 · doi:10.1016/0895-7177(91)90068-I [8] Saranen, J. (1991): The modified quadrature method for logarithmic-kernel integral equations on closed curves. J. Integral Equations Appl.3, 575-600 · Zbl 0747.65100 · doi:10.1216/jiea/1181075650 [9] Saranen, J., Sloan, I.H. (1991): Quadrature methods for logarithmic-kernel integral equations on closed curves. IMA J. Numer. Anal.12, 167-187 · Zbl 0751.65069 · doi:10.1093/imanum/12.2.167 [10] Sloan, I.H. (1988): A quadrature-based approach to improving the collocation method. Numer. Math.54, 41-56. · Zbl 0668.65111 · doi:10.1007/BF01403890 [11] Sloan, I.H. (1992): Error analysis of boundary integral methods. Acta Numerical1, 287-339 · Zbl 0765.65109 · doi:10.1017/S0962492900002294 [12] Sloan, I.H., Burn, B.J. (1992): An unconventional quadrature method for logarithmic-kernel integral equations of closed curves. J. Integral Equations Appl.4, 117-151 · Zbl 0760.65131 · doi:10.1216/jiea/1181075670 [13] Sloan, I.H., Wendland, W.L. (1989): A quadrature-based approach to improving the collocation method for splines of even degree. Zeit. Anal. Anwend.8, 362-376 · Zbl 0698.65071 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.