Langdon, S.; Graham, I. G. Boundary integral methods for singularly perturbed boundary problems. (English) Zbl 0978.65109 IMA J. Numer. Anal. 21, No. 1, 217-237 (2001). The authors consider boundary integral methods applied to the modified Helmholtz equation \(-\Delta u + \alpha^2 u =0\) with \(\alpha\) real and possibly large. The layer potentials have kernels which become highly peaked for large \(\alpha\), causing standard discretization schemes to fail. The authors propose a new discrete collocation method based on a sophisticated rescaling and product rules on graded meshes. This method has a robust convergence behaviour as \(\alpha \rightarrow \infty\), verified by some numerical tests. Reviewer: Gunther Schmidt (Berlin) Cited in 3 Documents MSC: 65N38 Boundary element methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Keywords:singular perturbations; boundary integral method; modified Helmholtz equation; graded meshes; collocation; numerical tests PDFBibTeX XMLCite \textit{S. Langdon} and \textit{I. G. Graham}, IMA J. Numer. Anal. 21, No. 1, 217--237 (2001; Zbl 0978.65109) Full Text: DOI