Algorithm for the new FM-BIEM

The algorithm for the new FMM is given as follows:

**Steps 1-3.**Same as the steps 1-3 in the algorithm described in 3.3.**Step 4.**Computation of the coefficients of the exponential expansion:Compute the coefficients of the exponential expansion at each cell using (4.36)-(4.49), taking the origin (

*O*) at the centroid of the cell.**Step 5.**Computation of the coefficients of the local expansion:We compute the coefficients of the local expansion of cells of level

*l*, starting from*l*=2 and increasing*l*. Now we consider a cell*C*and another cell*C*' which is contained in the interaction list of*C*. Considering the position of*C*' relative to*C*, we translate the coefficients of the exponential expansion via (4.50) and (4.51) as the centre of the exponential expansion is shifted from the centroid of*C*(*O*) to that of*C*' ( ) and then use appropriate formulae in (4.52)-(4.65) to convert the coefficients of the exponential expansion to the coefficients of the local expansion. After carrying out these conversions about all cells in the interaction list of*C*, we add them together via (4.66) and (4.67) to obtain the contribution from the interaction list of*C*to the coefficients of the local expansion. To these contributions we add the coefficients of the local expansion of the parent of*C*with the origin shifted from the centroid of the parent to that of*C*via (3.64) and (3.65) to obtain the coefficients of the local expansion associated with*C*.**Step 6.**Evaluation of the integral in (3.46):This step is the same as the step 5 in the algorithm described in 3.3.