有限元素法之固有值問題在多邊形鼓膜上之誤差分析

Abstract

This Helmholtz equation occurs frequently in dynamic meteorology. In classical physics it is the equation of the vibrating membrane. Time-harmonic wave propagation, either elastic waves or electromagnetic waves, is a common phenomenon that appears in many applications such as acoustic wave scattering from submarines, noise reduction in silencers and mufflers, earthquake wave propagation. First, the Helmholtz equation reduces to that of solving a generalized eigenvalue problem. In the chapter 2, it shows that the order of the error of eigenvalue is O(h^2) by the application of the spectral theory. In the latest chapter, the Helmholtz equation can be solved via finite element methods. In these numerical results, the eigenvalues are solved throughout a linear triangular element is recognized as approximation. The eigenvalue are solved throughout a quadratic triangular element is recognized as exact solution, since this numerical method has much higher rate of convergence. In addition, we compare the difference between linear triangular element and quadratic triangular element by refining the mesh.