Abstract: We derive a foundational equation in the virtual boundary method,and essential conditions that the virtual boundary must satisfy.It shows that the virtual boundary element method is rigorous.For complex real boundaries,such as a concave boundary,it is approximate because the requirements for virtual boundary can not be satisfied.Code for electron trace in a power electron gun is explored.It shows that the accuracy of the virtual boundary element method is better than that of the boundary element method as the distance between virtual boundary and real boundary is optimum.

Key words: virtual boundary element method, boundary element method, virtual boundary shape, optimum distance, power electron gun, numerical analysis