# A Polynomial Spectral Calculus for Analysis of DG Spectral Element Methods

## TL;DR

This paper introduces a polynomial spectral calculus based on Legendre-Gauss-Lobatto quadrature, simplifying the analysis of multidimensional discontinuous Galerkin spectral element methods.

## Contribution

It develops a new polynomial spectral calculus leveraging summation by parts, aiding in the analysis of DG spectral element methods.

## Key findings

- Simplifies multidimensional DG spectral element analysis
- Provides a new mathematical tool based on polynomial calculus
- Enhances understanding of spectral element method stability

## Abstract

We introduce a polynomial spectral calculus that follows from the summation by parts property of the Legendre-Gauss-Lobatto quadrature. We use the calculus to simplify the analysis of two multidimensional discontinuous Galerkin spectral element approximations.