Fast Evaluation of Singular BEM Integrals Based on Tensor Approximations
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Submission date: 21. Dec. 2010
published in: Numerische Mathematik, 121 (2012) 3, p. 433-460
DOI number (of the published article): 10.1007/s00211-011-0436-6
MSC-Numbers: 15A69, 65499, 65N38
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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 -Tucker format. The tensor approximation has to be done only once and allows us to evaluate interpolants in (dr(m + 1)) operations in the (CP) format, or (dk3 + dk(m + 1)) operations in the -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.