The ACA algorithm implemented into the direct BEM approach to reduce its numerical complexity

Abstract

The boundary element method (BEM) as a numerical method can be applied for high-resolution gravity field modelling. To obtain numerical solutions of “cm-level” accuracy, it requires very refined level of the disretization which leads to enormous memory requirements. An implementation of the Hierarchical Matrices (H-matrices) can significantly reduce a numerical complexity of the BEM approach. Here we present an implementation of the Adaptive Cross Approximation (ACA) algorithm into the direct BEM formulation applied for the global gravity field modelling. The ACA algorithm is based on a multilevel matrix-partitioning scheme of the rank-revealing LU decomposition, which uses a low rank of the submatrix belonging to two far groups of points. The algorithm performs a series of decompositions, which results in an approximation of the original submatrix using the product of two sparse matrices with low ranks. This approach can significantly reduce enormous memory requirements. Numerical experiments present efficiency of the ACA algorithm that can achieve a memory saving of 98% for the very refined meshes.