Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes

The discrete null-field equation method (DNFEM) was proposed based on the null-field equation (NFE) of Green’s representation formulation, where only disk domains were discussed. However, the study of the DNFEM for bounded simply connected domains S is essential for practical applications. Since the source nodes must be located outside of a solution domain S, the first issue in computations is how to locate them. It includes two topics—Topic I: The source nodes must be located not only outside S but also outside the exterior boundary layers. The width of the exterior boundary layers is derived as $O(1/N)$, where N is the number of unknowns in the DNFEM. Topic II: There are numerous locations for source nodes outside the exterior boundary layers. Based on the sensitivity index, several better choices of pseudo-boundaries are studied for bounded simply connected domains. The advanced study of Topics I and II needs stability and error analysis. The bounds of condition numbers (Cond) are derived for bounded simply connected domains, and they are similar to those of the method of fundamental solutions (MFS). New error bounds are also provided for bounded simply connected domains. The thorough study of determining better locations of source nodes is also valid for the MFS and the discrete boundary integral equation method (DBIEM). The development of algorithms based on the NFE lags far behind that of the traditional boundary element method (BEM). Some progress has been made by following the MFS, and reported in this paper. From the theory and computations in this paper, the DNFEM may become a competent boundary method in scientific/engineering computing.