Comparison of two sets of low-order basis functions for tetrahedral VIE modeling

A convergence study is made for the two types of low-order basis functions for the volume integral equation (VIE). Both functions impose the continuity of the normal component of the electric flux through the faces. The one basis function is that of Schaubert, Wilton, and Glisson (1984) and is face-based. Another basis function was first introduced by de Carvalho and de Souza Mendes (1999) and is edge-based. The exact number of unknowns for the edge-based functions is determined in this study. The study demonstrates a better performance of the edge-based basis functions compared to the face-based bases. First, the edge-based basis functions have nearly the same or a faster convergence rate for equal tetrahedral meshes. They also show a high numerical stability. Second, for the same tetrahedral mesh, the number of unknowns for the edge-based functions is considerably smaller. The ratio of unknowns (edge-based versus face-based) ranges from 0.6 for rough plate meshes to approximately 0.5 for large volumetric meshes. Third, the edge-based functions are piecewise constant and are easily implemented into the method of moments. Their disadvantage is a preliminary condition "operation," which implies the elimination of the space of the basis set.