Boundary integral methods for singularly perturbed boundary problems (Q2713125)

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.