Identifying Common Properties of Coefficient Matrices Appearing in Conventional or Invariant Scheme of Method of Fundamental Solutions

Abstract

In previous studies on the method of fundamental solutions applied to Dirichlet problems, the only method for verifying the regularity of coefficient matrix has been by calculating the determinant and checking if it is non-zero directly. These verifications were carried out in some specific cases. This paper presents an identification of common properties of coefficient matrices within the method of fundamental solutions related to regularity. It establishes sufficient conditions for coefficient matrices to be regular and diagonally dominant in the two-dimensional plane or the three-dimensional space. However, when dealing with both the conventional scheme and the invariant scheme, we encounter issues where achieving a good arrangement of charges while maintaining diagonal dominance is challenging. One advantage of the invariant scheme over the conventional scheme is found to be based on the solvability of the shift of the boundary data. © 2025 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.