ABSTRACT
This chapter studies an alternative derivation of the electrostatic finite element equations using the principle of minimum energy. It derives the function of the electric field and shows how the contribution of dielectric charges is automatically included. The chapter shows that the spatial distribution of electric field given by a solution of the Poisson equation corresponds to a state of minimum field energy integrated over the system volume. It reviews results from the calculus of variations that are necessary for the derivation. The minimum energy is applied to the fields of triangular elements in a two dimensional system. The electrostatic energy in elements surrounding a vertex depends on the electrostatic potential at the vertex and at neighboring points. The chapter summarizes the motivations for and procedures to derive finite-element equations from high-order polynomial expansions of the potential. This approach offers enhanced accuracy in applications where materials have uniform properties.
