The present paper deals with a new approach to the finite element approximation of an 'essentially three-dimensional' problem of eigenmodes of electromagnetic oscillation in resonator cavities. To this end, it is suggested to use specific basis vector functions whose forms are deduced from the properties of the corresponding variational formulation. It is shown that the algebraic eigenvalue problem thus obtained preserves basic properties of the original differential problem. The paper describes the general scheme and some specific realizations of the approach suggested.The problem of computing eigenmodes of electromagnetic oscillation in resonator cavities is of considerable importance, for example, in charged-particle accelerator design. The first stage of numerical solution of this problem consists in its discretization. In some cases, the original problem can be reduced to a two-dimensional scalar eigenvalue problem with the second-order differential elliptic operator, for example, if the cavity considered is a domain with a constant cross-section or a rotation body (see [5,7]). In this case, the discretization does not raise any difficulties as there exist known approximations of similar problems by finite difference and finite element methods, and by others.Let us consider now the case of 'essentially three-dimensional' problems. The technique of finite difference approximation of such problems suggested in [2, 3] is based on the results of [6] and exploits difference counterparts of basic differential operators, which preserve their properties and mutual relationships (also see [10]). However, this technique is applicable only to uniform orthogonal grids and, additionally, imposes substantial restrictions on the domain under consideration, i.e. the boundary of the domain has to belong to a union of grid surfaces. In many cases, these requirements are unacceptable. Thus, there arises a necessity to approximate 'essentially three-dimensional' problems by the finite element method.Attempts of this kind were made in [1,4,8,9] and in some other papers. However, the results cannot be considered to be completely satisfactory as the discrete problems obtained therein do not possess certain important properties inherent in the original differential problem. In addition, the use of the basis functions suggested raises difficulties involved in approximating boundary conditions. From the standpoint of the author, the above-mentioned shortcomings and difficulties are caused mainly by an inappropriate choice of basis functions of the finite element method, and in some cases, additionally, by an inappropriate choice of a variational functional. It can be shown that the basis functions used so far are redundantly smooth and thus 'unnatural' for the problem discussed. f Originally published in Russian in: Chislennye Melody i Matematicheskoe Modelirovanie (
