| Abstract | The evaluation of sums (matrix-vector products) of the solutions ofthe
three-dimensional biharmonic equation can be accelerated using the
fast
multipole method, while memory requirements can also be significantly
reduced. We develop a complete translation theory for these equations.
It is shown that translations of elementary solutions of the
biharmonic
equation can be achieved by considering the translation of a pair of
elementary solutions of the Laplace equations. Compared to previous
methods that required the translation of five Laplace elementary
solutions for the biharmonic Green's function, and much larger numbers
for higher order multipoles, our method is significantly more
efficient.
The theory is implemented and numerical tests presented that
demonstrate
the performance of the method for varying problem sizes and accuracy
requirements. In our implementation the FMM\ is faster than direct
solution for a matrix size of $550$ for an accuracy of $10^{-3},$ 950
for an accuracy of $10^{-6} and $N=3550$ for an accuracy of
$10^{-9}$.
|
