  This paper presents a product to sum approach for a fast and efficient matrix
filling in a hierarchical finite-element method (FEM). Due to the existence of
a coupling factor arising from the material and Jacobian inhomogeneities in
curved inhomogeneous elements, the calculation of the FEM matrix elements
should be carried out through full multidimensional integrations. This reduces
the efficiency of the higher order FEM solvers especially when the coupling
factor varies rapidly inside the elements. In the product to sum approach,
every product of the basis and weighting polynomials is replaced with a sum of
appropriate polynomials. This reduces the number of required multidimensional
integrals significantly and converts the integration into a summation.
Therefore, the method will be efficient if the number of summation terms in the
product to sum conversion is as small as possible.
