The spectral element method constructed by the $Q^k$ ($k\geq 2$) continuous finite element method with $(k+1)$-point Gauss-Lobatto quadrature on rectangular meshes is a popular high order scheme for solving wave equations in various applications. It can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is $(k+2)$-order accurate in discrete 2-norm for smooth solutions. The same proof can be extended to the spectral element method solving linear parabolic and Schr\"odinger equations. 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.

翻译：光谱元件方法由 $k$(k\geq 2$) 的连续限制元件方法构建,在矩形间贝上使用$(k+1) 点高斯- 洛巴托等离子体,这是解决各种应用中的波方方程的流行高顺序方案,也可以被视为所有高斯- 洛巴托点的有限差异方案。我们证明,这一有限差异方案在离散 2- 诺姆 中精确度为 $(k+2), 以平滑的解决方案。同样的证据可以扩大到光谱元素方法,解决线性抛物和 Schr\ " odinger 等式。主要结果也适用于可顺利绘制成单位方形角角角矩形的曲线光谱元素方法。