Hierarchical matrices and adaptive cross approximation applied to the boundary element method with multi-domain governed by iterative coupling

The approach presented in this paper is based on the Adaptive Cross Approximation (ACA) applied to the matrices coming from the Boundary Element Method (BEM) with multi-domain. The algorithm uses a hierarchical matrix (H-matrix) storage approach splitting the coefficient matrices representing the interactions inside the sub-domains into many blocks where rank of the off-diagonal blocks is reduced with the help of ACA approximation. The sub-domains are then coupled through the iterative process. These optimisations of the coefficient matrices in conjunction with highly effective algorithms for manipulation with H-matrices allow one to perform the operation of matrix-vector multiplication with almost linear complexity O(NlogN). The approach allows one to solve the linear systems of equations for BEM with multi-domain having nearly 100.000 DOFs using the usual PC. This paper formulates the approach and demonstrates its numerical properties by means of a theoretical example involving a cube with 27 sub-domains.