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In BIEM an integral equation is given by
is an unknown function on S, f is a
given function on S and K is a given kernel function on
When one solves the integral equation numerically one
discretises it into the form of a matrix-vector product.
In FM-BIEM an iterative method is used as a solver for linear
equations. Namely FMM is utilised to reduce the computational
complexity for the multiplication of a matrix and a
candidate vector in BIEM. Indeed, FMM can reduce this complexity from
O(N2) to O(N). In FM-BIEM the integral in (2.1) is
evaluated with direct computation in the same manner as in the
conventional BIEM only when
is in the neighbourhood of
because of a singularity at
of a kernel function.
Otherwise, the integral is evaluated efficiently with FMM.