# Pole condition for singular problems: the pseudospectral approximation

@article{Huang1993PoleCF, title={Pole condition for singular problems: the pseudospectral approximation}, author={Weizhang Huang and David M. Sloan}, journal={Journal of Computational Physics}, year={1993}, volume={107}, pages={254-261} }

Abstract This paper deals with the pseudospectral solution of differential equations with coordinate singularities such as those which describe situations in spherical or cylindrical geometries. We use the differential equation, together with a smoothness assumption on the solution, to construct "pole conditions." The pole conditions, which are straightforward and easily implemented, serve as numerical boundary conditions at the coordinate singularity. Standard pseudospectral methods, including…

## 65 Citations

Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation

- MathematicsSIAM J. Numer. Anal.
- 2003

This paper considers an existing approach, which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used Gauss--Lobatto collocation scheme.

A Pseudospectral Approach for Polar and Spherical Geometries

- GeologySIAM J. Sci. Comput.
- 1995

Viewing pseudospectral methods as a limiting case of finite difference methods (rather than as based on expansions in terms of orthogonal functions) leads naturally to very simple, yet highly effective, fast Fourier transform (FFT)-based pseudospecting methods in such geometries.

A rational spectral collocation method for third-order singularly perturbed problems

- MathematicsJ. Comput. Appl. Math.
- 2016

A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates

- Mathematics
- 2000

In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for…

Spectral Chebyshev-Fourier collocation for the Helmholtz and variable coefficient equations in a disk

- MathematicsJ. Comput. Phys.
- 2008

Application of the pseudo-spectral method to 2D eigenvalue problems in elasticity

- MathematicsNumerical Algorithms
- 2008

A pseudo-spectral approach to 2D vibrational problems arising in linear elasticity is considerede using differentiation matrices and it is shown that it is necessary to apply an additional "pole" condition to deal with ther=0 coordinate singularity arising in the case of a 2D disc.

A Fourier-Legendre spectral element method in polar coordinates

- MathematicsJ. Comput. Phys.
- 2012

Numerical Treatment of Polar Coordinate Singularities

- Mathematics
- 2000

The treatment of the geometrical singularity in cylindrical and spherical coordinates has for many years been a difficulty in the development of accurate finite difference (FD) and pseudo-spectral…