The Finite Element Method in Electromagnetics, 3rd Edition

CHAPTER 6VARIATIONAL PRINCIPLES FOR ELECTROMAGNETICS

As we saw in the preceding four chapters, the variational method is one of the two methods often employed to formulate finite element solutions. There are several advantages associated with the variational method. The primary advantage is its well-established solid foundation in physics and mathematics, which may permit a physical interpretation and provide better understanding of a physical problem. Another advantage is that through the variational procedure one can clearly demonstrate the differences between essential and natural conditions, which are difficult for a beginner to comprehend. Other advantages include convenience of description and elegance of formulation. Because of these, in the past the variational method was usually preferred over Galerkin’s method.

However, the variational method has several disadvantages that make it an unpopular choice now, particularly in electromagnetics. Unlike Galerkin’s method, which starts directly with differential equations, the variational method starts from a variational formulation. Therefore, applicability of the method depends directly on the availability of such a variational formulation. In the preceding chapters, the variational formulation was always stated first and then proved by taking the first variation of the functional. This is a widely adopted practice for a historical reason. Before the advent of the computer, it was difficult to solve a variational problem ...