Exact calculation of matrix entries in space-time Galerkin time domain boundary element method for the scalar wave equation

The contribution of this work is to propose an exact calculation for the interaction elements appearing in space-time time domain boundary element methods based on a dimensionality reduction procedure. Similar methods have been applied in the frequency domain, while in the time domain only numerical or semi-analytical approaches have been proposed. The calculation presented here replaces a four dimensional interaction integral with a sum of lower dimensional integrals of a new effective Green function that can be evaluated in closed form. In this paper the method proposed is applied to scalar basis an testing function on flat supports but it can be adapted to the electromagnetic case. The results can be used as a benchmark to evaluate the precision and stability of existing numerical methods. It allows to distinguish instabilities inherent to the method and those that appear as a consequence of limited quadrature precision.