Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods applied to boundary val ue problems for the positive definite Helmholtz-type problem -DeltaU + alph a U-2 = 0 in a bounded or unbounded domain, with the parameter alpha real a nd possibly large. Applications arise in the implementation of space-time b oundary integral methods for the heat equation, where alpha is proportional to 1/root deltat, and deltat is the time step. The corresponding layer pot entials arising from this problem depend nonlinearly on the parameter alpha and have kernels which become highly peaked as alpha --> infinity, causing standard discretization schemes to fail. We propose a new collocation meth od with a robust convergence rate as alpha --> infinity. Numerical experime nts on a model problem verify the theoretical results.