Shen, Jie Efficient spectral-Galerkin methods. III: Polar and cylindrical geometries. (English) Zbl 0890.65117 SIAM J. Sci. Comput. 18, No. 6, 1583-1604 (1997). The paper is the third in the series for developing efficient spectral-Galerkin methods for elliptic equations. [For part II see ibid. 16, No. 1, 74-87 (1995; Zbl 0840.65113).] The special form of the spectral-Galerkin method for second- and fourth-order equations in polar and cylindrical geometries is presented. The proposed method is extremely accurate and efficient.The methods in question are based on appropriate variational formulations which incorporate naturally the pole conditions. In particular, the computational complexities of the Chebyshev-Galerkin method in a disk and the Chebyshev-Legendre-Galerkin method in a disk or a cylinder are quasi-optimal, that is optimal up to a logarithmic term. As an indication of efficiency, the CPU time for the Poisson solver on a disk by the Chebyshev-Galerkin method is only about 70% of the corresponding finite difference code hwsplr.f routine in the known package FISHPACK of FORTRAN subprograms for the solution of elliptic partial differential equations. Reviewer: J.Vaníček (Praha) Cited in 2 ReviewsCited in 61 Documents MSC: 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35J40 Boundary value problems for higher-order elliptic equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:Legendre polynomial; Chebyshev polynomial; Poisson equation; spectral-Galerkin method; fourth-order equations; Chebyshev-Legendre-Galerkin method Citations:Zbl 0811.65097; Zbl 0840.65113 Software:FISHPAK; CGS; LAPACK; hwsplr.f PDFBibTeX XMLCite \textit{J. Shen}, SIAM J. Sci. Comput. 18, No. 6, 1583--1604 (1997; Zbl 0890.65117) Full Text: DOI