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Boundary integral methods for singularly perturbed boundary problems. (English) Zbl 0978.65109

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.

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
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