Localized singular boundary method for the simulation of large-scale problems of elliptic operators in complex geometries

Abstract

In this paper, the novel localized singular boundary method (LSBM) in conjunction with the Chebyshev collocation scheme (CCS) is developed to simulate large-scale inhomogeneous problems governed by elliptic operators. By means of the Gauss–Lobatto collocation points, the CCS is employed to obtain the particular solution. With the obtained particular solution, the inhomogeneous elliptic boundary value problem can be transformed into a classical homogeneous problem which can be solved by the LSBM. In the LSBM, the whole computational domain with distributed nodes is divided into a series of overlapping circle-shaped sub-domains. For each sub-domain, the moving least square method and the conventional singular boundary method (SBM) are utilized. Considering that the matrix generated by the CCS-LSBM is sparse, this scheme is granted the tremendous potential to address large-scale problems. Numerical experiments are presented to verify the effectiveness and accuracy of the proposed scheme for problems with mixed boundary conditions, sophisticated analytical solutions, and complex computational domains, respectively.