Masatoshi Ikeuchi

The alpha interpolation method for the solution of an eigenvalue problem

Abstract We present a weighted residual finite element method for the solution of an eigenvalue problem. As a test function, we take a linear combination of two functions which belong to different spaces. We call this method the alpha interpolation method (AIM) for the eigenvalue problem. We compare the AIM with the Standard-Galerkin finite element method (SGFEM).

Arbitrarily Shaped Hollow Waveguide Analysis by the

An efficient numerical method named the

\alpha

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-Interpolation Method

-interpolation method (AIM) has been developed for solving hollow waveguide problems. The AIM is a simpler and more systematic procedure than the finite-element method. And when the optimum interpolation parameter is chosen for any number of linear triangular elements, the cutoff wavenumbers or the exact eigenvalues may be estimated by the AIM. Numerical results and comparisons are given.

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

Abstract The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method. The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carres method, Kulstruds method and the stationary iterative method using Frankels theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions. As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.