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On the computation of multi-dimensional single layer harmonic potentials via approximate approximations. (English) Zbl 1072.65029

The authors suggest a new strategy for the computation of surface potentials in \(\mathbb{R}^d\) of the form \[ \int Q(p-q) f(q)\, d\sigma(q)\,, \] where \(f\) is the density and \(Q\) is the kernel, singular at the origin. The key step of the procedure is the approximation of \(f\) by quasi-interpolation by locally supported smooth radial basis functions centred at regularly distributed nodes on the surface (so-called “approximate approximation”). As the second step, the potentials of local basis functions are approximated by integrals over the tangential space using local parametrization of the surface at the centre of each basis function and taking asymptotic expansion of the potential. The authors show also how to choose the radial basis functions allowing reduction of \((d-1)\)-dimensional integrals to one-dimensional ones. The case of the single-layer harmonic potential (with \(Q(p)=|p|^{2-d}\)) is considered in detail, although extensions to other kernels are also discussed. These results are illustrated with some numerical examples.

MSC:

65D30 Numerical integration
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