A new approach to multi-domain fast multipole boundary element method

The fast multipole boundary element method (FMBEM) is a powerful technique for solving large-scale problems. Its effectiveness heavily relies on the iterative solver, which in turn depends crucially on the performance of the preconditioner. Although a leaf -based preconditioner has proven effective in the single domain FMBEM (SFB), it encounters challenges in the multi -domain FMBEM (MFB). To overcome this challenge, this paper proposes a cell renumbering algorithm to construct a leaf -based preconditioner for MFB. This algorithm renumbers the cells in the tree structure, taking into account the influence of the tree structure's traversal order on the coefficient matrix distribution. This renumbering enables the generation of a leaf -based preconditioner matrix for MFB, with specialized processing for sub -matrix blocks associated with the interface, accommodating both aligned and misaligned cells on both sides of the interface. In cases where the tree structure is not aligned, each interface -related submatrix may correspond to two or more leaves, resulting in a larger matrix size and increased computational cost for preconditioner calculations. To mitigate this computational burden, this paper proposes an improved cell renumbering scheme, which directly manipulates the index set to align the cell indices within the tree structure, thereby reducing computational costs. The utilization of this renumbered preconditioner in MFB not only achieves fast convergence of iterative solvers but also retains the advantages as in SFB. Through these techniques, this paper proposes a new approach to MFB. Numerical results presented in this paper demonstrate the effectiveness and universality of the proposed cell renumbering algorithm and preconditioner for MFB.