Application Of Higher-order Hierarchical Vector Finite Elements For Solving The Problems Of Resonant Cavities

In recent years, lower-order vector finite elements methods have not been satisfied withthe development of higher accuration of the analysis of the electromagnetic engineering. Inthis situation, higher-order basis functions have been presented by some scholars. There aretwo types of higher-order vector elements. The first type is the interpolatory basis functions;in this case, the interpolatory basis functions are defined on a set of points on the element; thesecond approach is hierarchal basis functions. Higher-order basis functions are formed byadding new functions to the lower-order basis functions. The advantage of these basisfunctions is that they permit the use of different orders in a problem. For mesh refinementpurposes, hierarchal elements are very useful.In this paper we analysis the eigenvalue problems of waveguides and cavities usinghigher-order hierarchical vector basis functions based on tetrahedral finite elements. Firstly,we discuss the basic steps of vector finite elements including discretization of domain,selection of the interpolation functions, evaluation of elemental matrices and solution of thesystem of equations. Then, the formulation of the higher-order hierarchal vector basisfunctions has been presented. Last of all, the numerical results of several examples show thatusing the higher-order basis functions can reduce unknowns by larger discretization size toobtain the same accuracy with the lower-order basis functions, and the numerical solution canapproach to the analytical solution more quickly with mesh refinement.