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. 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 H-Tucker format. The tensor approximation has to be done only once and allows us to evaluate interpolants in O(dr(m+ 1)) operations in the (CP) format, or O(dk3+ dk(m+1)) operations in the 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.