A sparse spectral method on triangles
For researchers in computational PDEs, this provides a practical and efficient spectral method for triangular domains, though it is an incremental extension of existing univariate techniques.
The paper develops a sparse spectral method for solving linear PDEs on triangles using bivariate orthogonal polynomials, enabling rapid solutions with polynomial degrees in the thousands and sparse discretizations with millions of degrees of freedom.
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.