# A sparse spectral method on triangles

## TL;DR

This paper develops a spectral method using bivariate orthogonal polynomials on triangles, enabling efficient, sparse discretizations for solving large-scale linear PDEs with high polynomial degrees.

## Contribution

It introduces a spectral method on triangles leveraging analogues of univariate tools, allowing for sparse, high-degree polynomial discretizations of PDEs.

## Key findings

- Enables solving PDEs with polynomial degrees in the thousands.
- Achieves sparse discretizations with millions of degrees of freedom.
- Provides practical algorithms for spectral methods on triangular domains.

## Abstract

In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators. This allows us to derive a practical spectral method for solving linear partial differential equations on triangles with sparse discretizations. We can thereby rapidly solve partial differential equations using polynomials with degrees in the thousands, resulting in sparse discretizations with as many as several million degrees of freedom.