×

zbMATH — the first resource for mathematics

Efficient spectral-Galerkin methods for polar and cylindrical geometries. (English) Zbl 1180.65160
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 inhomgeneous boundary conditions is also addressed.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Auteri, F.; Quartapelle, L., Galerkin spectral method for the vorticity and stream function equations, J. comput. phys., 149, 2, 306-332, (1999) · Zbl 0934.76065
[2] Bernardi, C.; Dauge, M.; Maday, Y., Spectral methods for axisymmetric domains, Series in applied mathematics (Paris), vol. 3, (1999), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier Paris
[3] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Publications Inc. Mineola, NY · Zbl 0987.65122
[4] Chen, H.; Su, Y.; Shizgal, B.D., A direct spectral collocation Poisson solver in polar and cylindrical coordinates, J. comput. phys., 160, 2, 453-469, (2000) · Zbl 0951.65125
[5] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. algorithms, 42, 2, 137-164, (2006) · Zbl 1103.65119
[6] Eisen, H.; Heinrichs, W.; Witsch, K., Spectral collocation methods and polar coordinate singularities, J. comput. phys., 96, 2, 241-257, (1991) · Zbl 0731.65095
[7] Fornberg, B., A pseudospectral approach for polar and spherical geometries, SIAM J. sci. comput., 16, 5, 1071-1081, (1995) · Zbl 0831.65119
[8] Guo, B.; Shen, J.; Wang, L.-L., Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. sci. comput., 27, 305-322, (2006) · Zbl 1102.76047
[9] Guo, B.; Wang, L.-L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. approx. theory, 128, 1, 1-41, (2004) · Zbl 1057.41003
[10] Heinrichs, W., Spectral collocation schemes on the unit disc, J. comput. phys., 199, 1, 66-86, (2004) · Zbl 1057.65089
[11] Huang, W.; Ma, H.; Sun, W., Convergence analysis of spectral collocation methods for a singular differential equation, SIAM J. numer. anal., 41, 6, 2333-2349, (2003) · Zbl 1058.65076
[12] Huang, W.; Sloan, D.M., Pole condition for singular problems: the pseudospectral approximation, J. comput. phys., 107, 2, 254-261, (1993) · Zbl 0785.65091
[13] Lynch, R.E.; Rice, J.R.; Thomas, D.H., Direct solution of partial difference equations by tensor product methods, Numer. math., 6, 185-199, (1964) · Zbl 0126.12703
[14] Matsushima, T.; Marcus, P.S., A spectral method for polar coordinates, J. comput. phys., 120, 2, 365-374, (1995) · Zbl 0842.65051
[15] Shen, J., Efficient spectral-Galerkin method. I. direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. sci. comput., 15, 6, 1489-1505, (1994) · Zbl 0811.65097
[16] Shen, J., Efficient spectral-Galerkin methods. III. polar and cylindrical geometries, SIAM J. sci. comput., 18, 6, 1583-1604, (1997) · Zbl 0890.65117
[17] Shen, J., A new fast chebyshev – fourier algorithm for Poisson-type equations in polar geometries, Appl. numer. math., 33, 183-190, (2000) · Zbl 0967.65108
[18] Verkley, W.T.M., A spectral model for two-dimensional incompressible fluid flow in a circular basin. I. mathematical formulation, J. comput. phys., 136, 1, 100-114, (1997) · Zbl 0889.76071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.