5. The Vector Finite Element Method

Originally only the so called node finite element analysis has been used. By the node finite elements the unknown field values are assigned to the element vertexes. It is obvious that vector field quantities can be described with their components on the vertexes. Thus node elements can be used also for vector field calculations. In this case two or three scalars must be assigned to each vertex for the two- or three-dimensional case, respectively. Unfortunately using the coordinates of the vector gives problems. For example, the continuity requirements for the approximated vector fields cannot be easily fulfilled and also applying of the boundary conditions is very cumbersome [55]. To overcome these obstacles the so-called edge elements are used by the vector finite element analysis [56]. They are assigned to the edges of an element and are excellently suited for numerical solution of partial differential equations for the field intensities derived from the Maxwell equations . However the popular believe, that edge elements eliminate the problems with the so-called spurious modes is erroneous. Spurious solutions are avoided only by a proper finite element formulation [57].

In Chapter 4 at first the two-dimensional case is discussed, where the finite element assembling mechanism is explained in detail. In the three-dimensional case the method works analogously, actually only the element matrix is different. There is also no difference in the way of assembling between scalar and vector finite element analysis. Using the vector elements the assembling is performed in the same manner, namely element wise. The building of the element matrix is different and is discussed comprehensively. Since it is no longer necessary to explain the assembling process in detail, this chapter handles the three-dimensional case before the two-dimensional one. It starts with some typical applications for the edge elements.

Furthermore, it is assumed that the constitutive parameters are time invariant and scalars ( $\utilde{\epsilon}{\rightarrow}\epsilon$ , $\utilde{\mu}{\rightarrow}\mu$ , and $\utilde{\gamma}{\rightarrow}\gamma$ ). If it is required that the constitutive parameters are tensors, just replace for example, $(\epsilon\vec{E})$ by $(\utilde{\epsilon}{\cdot}\vec{E})$ , $(\epsilon\vec{\nabla}\varphi)$ by $(\utilde{\epsilon}{\cdot}\vec{\nabla}\varphi)$ , or $[a \vec{\nabla}{\times}\vec{b}]$ by $[\utilde{a}{\cdot}(\vec{\nabla}{\times}\vec{b})]$ .