Efficient spectral-Galerkin methods for polar and cylindrical geometries

Authors:
Y. -Y. Kwan
Affiliations:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA
Venue:
Applied Numerical Mathematics
Year:
2009

Citing 15
Cited 4

Spectral collocation methods and polar coordinate singularities

Journal of Computational Physics
Pole condition for singular problems: the pseudospectral approximation

Journal of Computational Physics
Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials

SIAM Journal on Scientific Computing
A pseudospectral approach for polar and spherical geometries

SIAM Journal on Scientific Computing
A spectral method for polar coordinates

Journal of Computational Physics
A spectral model for two-dimensional incompressible fluid flow in a circular basin I. Mathematical formulation

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Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries

SIAM Journal on Scientific Computing
Galerkin spectral method for the vorticity and stream function equations

Journal of Computational Physics
A new fast Chebyshev—Fourier algorithm for Poisson-type equations in polar geometries

Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A direct spectral collocation Poisson solver in polar and cylindrical coordinates

Journal of Computational Physics
Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical Polynomials

SIAM Journal on Scientific Computing
Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation

SIAM Journal on Numerical Analysis
Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces

Journal of Approximation Theory
Spectral collocation schemes on the unit disc

Journal of Computational Physics
Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

Journal of Scientific Computing

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier Series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions

Journal of Computational Physics
Matrix decomposition algorithms for elliptic boundary value problems: a survey

Numerical Algorithms
A Fourier-Legendre spectral element method in polar coordinates

Journal of Computational Physics
A sparse mesh for Compact Finite Difference-Fourier solvers with radius-dependent spectral resolution in circular domains

Computers & Mathematics with Applications

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Abstract

A new spectral-Galerkin approach for solving the Poisson-type equation in polar geometry is introduced and analyzed. The pole singularity is treated naturally through an appropriate variational formulation. Clustering of collocation points near the pole, a problem common to the spectral-Galerkin algorithms in the literature, is prevented through a change of variable in the radial direction. The method is very efficient and gives spectral accuracy, and can be easily adopted to solve problems in cylindrical geometries and with general boundary conditions. Boundary lifting of general inhomogeneous boundary conditions is also addressed.