A MESHLESS COLLOCATION METHOD BASED ON HIERARCHICAL MATRICES FOR EFFICIENT NUMERICAL ANALYSIS OF LARGE-SCALE LINEAR SYSTEMS

This paper presents a novel approach to solving large-scale linear algebra problems by integrating hierarchical matrices (H-matrices) with a semi-analytic meshless collocation method (singular boundary method).The singular boundary method eliminates the need for mesh generation by using a set of discrete nodes to approximate the solution of partial differential equations.However, this approach often leads to dense linear systems that are computationally expensive and memory-intensive.To address this challenge, we employ H-matrices, which decompose these dense matrices into hierarchical sub-blocks that can often be approximated by low-rank matrices.This decomposition significantly reduces computational complexity from the conventional to or lower, depending on the matrix structure.The implementation of these data-sparse representations significantly enhances the efficiency of numerical computations by reducing both complexity and storage requirements traditionally associated with matrix operations.Our results demonstrate that the combination of Hmatrices with the singular boundary method not only reduces memory requirements but also accelerates matrix operations, such as matrix-vector multiplication and inversion.The validity of the proposed method for solving the Laplace equation and the Helmholtz equation is verified by benchmark examples, highlighting its potential for large-scale engineering problems.