Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems

A generalized theory of higher order vector finite elements is presented. The methodology, that is introduced, is used for the construction of new second and third order generalizations of the concept of tetrahedral edge elements. Basic theoretical issues about their construction, such as the proper choice and placement of degrees of freedom and the correct modeling of irrotational fields are discussed. The presented theory and methodology are also extended to the hexahedral element case. Some practical problems and difficulties related to the numerical implementation of higher order vector finite elements are also dealt with. Finally, both second and third order elements are successfully implemented in several electromagnetic field or potential formulations.