Accuracy of spectral element method for wave, parabolic and Schr\"{o}dinger equations

TL;DR

This paper proves that the spectral element method using Gauss-Lobatto quadrature achieves high-order accuracy for wave, parabolic, and Schrödinger equations, including on curvilinear meshes, enhancing its theoretical understanding.

Contribution

It provides a rigorous proof of the $(k+2)$-order accuracy of the spectral element method for multiple PDEs, extending previous results to curved meshes and different equation types.

Findings

Spectral element method is $(k+2)$-order accurate for smooth solutions.

The proof extends to parabolic and Schrödinger equations.

Results apply to curved meshes mapped from rectangular ones.

Abstract

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.