R-functions in the generalized method of eigenoscillations

The generalized method of eigenoscillations (GME), as applied to solving a wide class of internal and external diffraction problems, is a further development of the method of eigenfrequencies. Some modifications of the GME are known, namely, the k-method (the eigenfrequency method), the e-method (eigenvalue in the equation), the w-method (eigenvalues in the impedance boundary condition), the p-method (eigenvalues in the conjugation condition), and the s-method (eigenvalues in the infinity condition). From the point of view of numerical analysis, the GME is based on a solution of some auxiliary eigenvalue problem in a complex-shaped domain. Here, the unknown solution is expanded in a series with respect to a proper system of basis functions, whose undetermined coefficients are found by one of variational or projective methods (Ritz, Bubnov-Galerkin, the least squares, etc.). The basis functions should satisfy definite requirements, in particularly, boundary conditions of the original problem (the latter requirement is obligatory for the Dirichlet boundary conditions). The choice of such functions is a complicated problem in the case of an arbitrarily shaped domain and it can be efficiently solved with the help of the theory of R-functions. In the report, the procedure of the combined use of the GME and the R-function method (RFM) is described and results of numerical experiments are presented.