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IMA Journal of Numerical Analysis 2001 21(1):217-237; doi:10.1093/imanum/21.1.217
© 2001 by Institute of Mathematics and its Applications
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Boundary integral methods for singularly perturbed boundary value problems

S. Langdon1 and I. G. Graham1

1 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK, e-mail: stephen.langdon{at}durham.ac.uk e-mail: igg{at}maths.bath.ac.uk

In this paper we consider boundary integral methods applied to boundary value problems for the positive definite Helmholtz-type problem –{Delta}U + {alpha}2U = 0 in a bounded or unbounded domain, with the parameter {alpha} real and possibly large. Applications arise in the implementation of space–time boundary integral methods for the heat equation, where {alpha} is proportional to 1/({surd}{delta}t), and {delta}t is the time step. The corresponding layer potentials arising from this problem depend nonlinearly on the parameter {alpha} and have kernels which become highly peaked as {alpha} -> {infty}, causing standard discretization schemes to fail. We propose a new collocation method with a robust convergence rate as {alpha} -> {infty}. Numerical experiments on a model problem verify the theoretical results.

Key Words: singular perturbation, boundary integral method, Helmholtz equation, heat equation, collocation, trigonometric polynomial

Received 11 August 1998. Accepted 12 December 1999.


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