# Sparse spectral methods for partial differential equations on spherical caps

no code implementations21 Dec 2020,

In recent years, sparse spectral methods for solving partial differential equations have been derived using hierarchies of classical orthogonal polynomials on intervals, disks, disk-slices and triangles.

Numerical Analysis Numerical Analysis 65N35

# Orthogonal polynomials in and on a quadratic surface of revolution

1 code implementation25 Jun 2019,

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids.

Classical Analysis and ODEs Numerical Analysis Numerical Analysis 42C05, 42C10, 65D15, 65D32

17

# A sparse spectral method on triangles

1 code implementation13 Feb 2019, ,

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.

Numerical Analysis 65N35

17

# Tensor calculus in polar coordinates using Jacobi polynomials

In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk.

Numerical Analysis Instrumentation and Methods for Astrophysics

17

# A fast and well-conditioned spectral method for singular integral equations

The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution.

Numerical Analysis 65N35, 65R20, 33C45, 31A10

62

# Fast computation of Gauss quadrature nodes and weights on the whole real line

The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial.

Numerical Analysis

299

# On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods

We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods.

Numerical Analysis 30C30, 41A30, 65D30, 65L10

7
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