Fast evaluation of singular BEM integrals based on tensor approximations

In this paper, we propose a method for the fast evaluation of integrals stemming from boundary element methods including discretisations of the classical single and double layer potential operators. Our method is based on the parametrisation of boundary elements in terms of a d-dimensional parameter tuple. We interpret the integral as a real-valued function f depending on d parameters and show that f is smooth in a d-dimensional box. A standard interpolation of f by polynomials leads to a d-dimensional tensor which is given by the values of f at the interpolation points. This tensor may be approximated in a low rank tensor format like the (CP) format or the ${{mathcal {H}}}$ -Tucker format. The tensor approximation has to be done only once and allows us to evaluate interpolants in ${{mathcal{O}}(dr(m+1))}$ operations in the (CP) format, or ${{mathcal{O}}(dk^3+dk(m+1))}$ operations in the ${{mathcal H}}$ -Tucker format, where m denotes the interpolation order and the ranks r, k are small integers. We demonstrate that highly accurate integral values can be obtained at very moderate costs.