A High Accuracy Spectral Element Method for Solving Eigenvalue Problems

High-Precision Chebyshev Spectral to Approximate Laplace Eigenvalue Problem

“…Based on the above discrete results analysis, this paper also compared the numerical results with the linear finite element. We compared error of the first five eigenvalues with the literature [12], in order to reduces the rounding error of the difference between the exact solution and the numerical solution when calculating the error. So we can simultaneously multiply 2 4 π to the exact solution and the numerical solution.…”

“…It can be seen from the figure that the error can eventually be reached 13 -10 , with the increase of the interpolation point, the accuracy of the approximation tends to be stable, and it can be basically maintained at about this amplitude. Compared with trigonal spectral elements, this method does not require triangular mesh planning [12], which can reduce the calculation amount and improve the calculation. The efficiency is basically the same from the solution accuracy and the convergence speed.…”

“…The boundary of this equation belongs to the type of aperiodic boundary; 3) According to the current experience, the first few eigenvalues are the objects that are the focus of attention. This example mainly calculates the first five eigenvalues and compared with the finite linear element method (Literature [12]). The Laplace equation's analytic solution is…”

“…Based on three types of special triangles (equal, isosceles right angles, and 30°, 60°, 90° triangular), the approximation formula for the eigenvalues in any triangular domain is given, reflecting the effect of a specific geometric region on the characteristic problem. Shan W, Li H proposed a spectral method based on an arbitrary triangle to solve the Lapalce eigenvalue problem in 2015 [12]. And then, Plestenjak B et al, studied the spectral collocation method to solve the multi-parameter eigenvalue problem [13], the application scope has been extended.…”