**Step 1.**Discretisation:Discretise

*S*in the same manner as in the conventional BIEM using boundary elements.**Step 2.**Construction of quad-tree structure:Consider a square which circumscribes

*S*and call this square the cell of level 0. Now take a cell (a parent cell) of level*l*and divide it into 4 equal sub squares whose edge length is half of that of the parent cell and call any of them a cell (a child cell) of level*l*+1 if some boundary elements belong to this square (See Fig.2.6). In this statement a boundary element is said to belong to a cell*C*if the centroid of the element is in*C*(See Fig.2.7). Stop the cell subdivision if the number of boundary elements belonging to the cell is smaller than a given number. A childless cell is called a leaf.**Step 3.**Computation of the multipole moments (Upward):First we compute the multipole moments associated with leaves using (2.4). Here the multipole moments associated with a cell

*C*stand for the integrals in (2.4) with*O*taken as the centroid of*C*. For a non-leaf cell*C*of level*l*, we compute the multipole moments associated with*C*by adding all the multipole moments of*C*'s children after shifting the origin from the centroids of*C*'s children to that of*C*via (2.7) (See Fig.2.8). This procedure is repeated for tracing the tree structure of cells upward (decreasing*l*).**Step 4.**Computation of the coefficients of the local expansion (Downward):We have to prepare some definitions first. We say that two cells are ``adjacent cells at level

*l*'' if these cells are both of the level*l*and share at least one vertex. Two cells are said to be ``well-separated at level*l*'' if they are not adjacent at level*l*but their parent cells are adjacent at level*l*-1. The list of all the well-separated cells from a level*l*cell*C*is called the interaction list of*C*. The maximum number of well-separated cells is 6^{2}-3^{2}=27 (See Fig.2.9). Now we compute the local expansion associated with a cell*C*. The local expansion associated with*C*represents a sum of the contributions due to boundary elements in cells of the interaction list of*C*and the contributions due to all boundary elements in cells which are not adjacent to*C*'s parent. The former is computed as we substitute the multipole moments associated with cells in the interaction list of*C*into (2.10) and the latter by translating the local expansion associated with*C*'s parent via (2.12) as the centre of the local expansion is shifted from the centroid of*C*'s parent ( ) to the centroid of*C*( ) (See Fig.2.8). The sum of these contributions gives the local expansion associated with*C*. We repeat these procedures starting from*l*=2 and increasing*l*along the tree structure until we reach leaves. When*l*=2, the coefficients of the local expansion are computed only by using (2.10). This is because cells of level 1 have no well-separated cells.**Step 5.**Evaluation of the integral in (2.1):We now compute the right-hand side of (2.1). Let

*C*be a cell of the level*l*to which the collocation point belongs, and let*C*' be a cell of level*l*adjacent to*C*. We compute the contribution from boundary elements in*C*' to the integral in (2.1) at collocation points in*C*directly if either*C*or*C*' is a leaf. When*C*is a leaf, we also compute the right-hand side of (2.9) using the coefficients of the local expansion associated with*C*. The sum of all these terms as*l*increases from 2 gives the integral in (2.1) evaluated at the collocation points of all boundary elements.**Step. 6**Update the candidate vector and go back to**Step. 3**.