부산대학교 도서관

This paper analyzes the accuracy of solving wave, hyperbolic and Schrödinger equations by the spectral element method. Firstly, the authors demonstrate that the spectral scheme constructed by the $Q^k$ continuous finite element method with $k+1$-point Gauss-Lobatto quadrature on rectangular meshes for solving wave equations is $k+2$-order accurate in the discrete 2-norm for smooth solutions. The same proof can be extended to the spectral element method for solving linear parabolic and Schrödinger equations. Moreover, the main result also applies to the spectral element method on curvilinear meshes that can be smoothly mapped to rectangular meshes on the unit square. \par Compared with other numerical methods, the high accuracy of the spectral element method has obvious advantages. However, the algorithmic principle of the spectral element method restricts the computational efficiency to some extent. The study of the error estimation for linear PDEs is meaningful; especially if it can be extended to nonlinear equations.