zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials. (English) Zbl 1152.65112
The authors consider some 1D and 2D fourth order differential equations supplied with first order (clamped) homogeneous and nonhomogeneous boundary conditions. They solve these problems by a Galerkin type method based on finite dimensional trial and test spaces spanned on Jacobi polynomials. In fact, they introduce two distinct bases of shape functions trying to minimize the bandwidth and condition number of the coefficient matrices. These special structured discretization matrices can also be efficiently inverted. Two fairly relevant examples are carried out.

MSC:
65N35Spectral, collocation and related methods (BVP of PDE)
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35J40Higher order elliptic equations, boundary value problems
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
WorldCat.org
Full Text: DOI
References:
[1] W.M. Abd-Elhameed, Spectral Galerkin method for solving second and fourth order differential equations by using ultraspherical polynomials, Cairo University, Egypt. M.Sc. Thesis, 2000
[2] Alperto, B. K.; Rokhlin, V.: A fast algorithm for the evaluation of Legendre expansions. SIAM J. Sci. statist. Comput. 12, 158-179 (1991) · Zbl 0726.65018
[3] Babuška, I.; Guo, B. Q.: Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions. Numer. math. 85, 219-255 (2000) · Zbl 0970.65117
[4] Babuška, I.; Guo, B. Q.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces, I. Approximability of functions in the weighted Besov spaces. SIAM J. Numer. anal. 39, 1512-1538 (2002) · Zbl 1008.65078
[5] Bernardi, C.; Maday, Y.: Spectral methods for the approximation of fourth-order problems: applications to the Stokes and Navier -- Stokes equations. Comput. & structures 30, 205-216 (1988) · Zbl 0668.76038
[6] C. Bernardi, Y. Maday, Some spectral approximations of one-dimensional fourth-order problems, ICASE Report No. 89 -- 36, ICASE, NASA-Langley Research Center, Hampton, VA, 1989
[7] Bernardi, C.; Maday, Y.: Some spectral approximations of fourth-order problems. Progress in approximation theory, 43-116 (1991)
[8] Bialecki, B.; Karageorghis, A.: A Legendre spectral collocation method for the biharmonic Dirichlet problem. Math. model. Numer. anal. 34, 637-662 (2000) · Zbl 0984.65121
[9] Bialecki, B.; Karageorghis, A.: A Legendre spectral Galerkin method for the biharmonic Dirichlet problem. SIAM J. Sci. comput. 22, 1549-1569 (2000) · Zbl 0986.65115
[10] Bjørstad, P.: Fast numerical solution of the biharmonic Dirichlet problem on rectangles. SIAM J. Numer. anal. 20, 59-71 (1983) · Zbl 0561.65077
[11] Bjørstad, P. E.; Tjøstheim, B. P.: Efficient algorithms for solving a fourth-order equation with the spectral-Galerkin method. SIAM J. Sci. comput. 18, 621-632 (1997) · Zbl 0939.65129
[12] Boyd, J. P.: Chebyshev and Fourier spectral methods. (2001) · Zbl 0994.65128
[13] Buzbee, B. L.; Dorr, F. W.: The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular domains. SIAM J. Numer. anal. 8, 722-736 (1971) · Zbl 0231.65083
[14] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamics. (1989) · Zbl 0717.76004
[15] Ciarlet, P. G.; Raviart, P. -A.: A mixed finite element method for the biharmonic equation. Symposium on mathematical aspects of finite elements in partial differential equations, 125-143 (1974) · Zbl 0337.65058
[16] Coutsias, E. A.; Hagstrom, T.; Torres, D.: An efficient spectral method for ordinary differential equations with rational function. Math. comp. 65, 611-635 (1996) · Zbl 0846.65037
[17] Doha, E. H.: The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. J. comput. Math. appl. 21, 115-122 (1991) · Zbl 0723.33008
[18] Doha, E. H.: The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function. J. comput. Appl. math. 89, 53-72 (1998) · Zbl 0909.33007
[19] Doha, E. H.: On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J. phys. A: math. Gen. 35, 3467-3478 (2002) · Zbl 0997.33004
[20] Doha, E. H.: Explicit formulae for the coefficients of integrated expansions for Jacobi polynomials and their integrals. Integral transforms and special functions 14, 69-86 (2003) · Zbl 1051.33004
[21] Doha, E. H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. phys. A: math. Gen. 37, 657-675 (2004) · Zbl 1055.33007
[22] Doha, E. H.; Bhrawy, A. H.: Efficient spectral-Galerkin algorithms for direct solution of second-order differential equations using Jacobi polynomials. Numer. algorithms 42, 137-164 (2006) · Zbl 1103.65119
[23] Funaro, D.: Polynomial approximations of differential equations. (1992) · Zbl 0774.41010
[24] Funaro, D.; Heinrichs, W.: Some results about the pseudospectral approximation of one dimensional fourth-order problems. Numer. math. 58, 399-418 (1990) · Zbl 0714.65074
[25] Gottlieb, D.; Orszag, S. A.: Numerical analysis of spectral methods: theory and applications. (1977) · Zbl 0412.65058
[26] Graham, A.: Kronecker products and matrix calculus: with applications. (1981) · Zbl 0497.26005
[27] Guo, B. -Y.: Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J. math. Anal. appl. 243, 373-408 (2000) · Zbl 0951.41006
[28] Guo, B. -Y.; Shen, J.; Wang, Z. -Q.: Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Internat. J. Numer. methods engrg. 53, 65-84 (2002) · Zbl 1001.65129
[29] Guo, B. -Y.; Wang, L. -I.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. comput. Math. 14, 227-276 (2001) · Zbl 0984.41004
[30] Heinrichs, W.: Line relaxation for spectral multigrid methods. J. comput. Phys. 77, 166-182 (1988) · Zbl 0649.65055
[31] Heinrichs, W.: Collocation and full multigrid methods. Appl. math. Comput. 26, 35-43 (1988) · Zbl 0637.65114
[32] Junghanns, V. P.: Uniform convergence of approximate methods for Cauchy type singular equation over ( - 1,1). Wiss. Z. Tech. hocsch. Karl-Mars stadt 26, 250-256 (1984) · Zbl 0575.65136
[33] Karageorghis, A.; Phillips, T. N.: Efficient direct methods for solving the spectral collocation equations for Stokes flow in rectangularly decomposable domains. SIAM J. Sci. statist. Comput. 10, 89-103 (1989) · Zbl 0665.76039
[34] Karageorghis, A.; Phillips, T. N.: Spectral collocation methods for Stokes flow in contraction geometries and unbounded domains. J. comput. Phys. 80, 314-330 (1989) · Zbl 0659.76037
[35] D.B. Knudson, A piecewise Hermite bicubic finite element Galerkin method for the biharmonic Dirichlet problem, Ph.D. thesis, Colorado School of Mines, Golden, CO, 1997
[36] Luke, Y.: The special functions and their approximations, 2. (1969) · Zbl 0193.01701
[37] Luke, Y.: Mathematical functions and their approximations. (1975) · Zbl 0318.33001
[38] Maday, Y.; Métivet, B.: Chebyshev spectral approximation of Navier -- Stokes equations in a two dimensional domain. Model. math. Anal. numer. 21, 93-123 (1986) · Zbl 0607.76024
[39] Malek, A.; Phillips, T. N.: Pseudospectral collocation methods for fourth order differential equations. IMA J. Numer. anal. 15, 523-553 (1995) · Zbl 0855.65088
[40] Shen, J.: Efficient spectral-Galerkin method I: Direct solvers of second-and fourth-order equations using Legendre polynomials. SIAM J. Sci. comput. 15, 1489-1505 (1994) · Zbl 0811.65097
[41] Shen, J.: Efficient spectral-Galerkin method II: Direct solvers of second-and fourth-order equations using Chebyshev polynomials. SIAM J. Sci. comput. 16, 74-87 (1995) · Zbl 0840.65113
[42] Shen, J.: Efficient Chebyshev Legendre Galerkin methods for elliptic problems. Houston J. Math., 233-240 (1996)
[43] Sivashinsky, G.: Nonlinear analysis of hydrodynamic instability in laminar flames, part I: Derivation of basic equations. Acta astronaut. 4, 1177-1206 (1977) · Zbl 0427.76047
[44] Stephan, E. P.; Suri, M.: On the convergence of the p-version of the boundary element Galerkin method. Math. comp. 52, 31-48 (1989) · Zbl 0661.65118
[45] Szegö, G.: Orthogonal polynomials. Amer. math. Soc. colloq. Publ. 23 (1985) · Zbl 65.0278.03
[46] Voigt, R. G.; Gottlieb, D.; Hussaini, M. Y.: Spectral methods for partial differential equations. (1984) · Zbl 0534.00017
[47] Wang, Z. -Q.; Guo, B. -Y.: A rational approximation and its applications to nonlinear partial differential equations on the whole line. J. math. Anal. appl. 274, 374-403 (2002) · Zbl 1121.41303
[48] Watson, G. N.: A note on generalized hypergeometric series. Proc. London math. Soc. (2) 23, xiii-xv (1925) · Zbl 51.0283.04
[49] Zang, T. A.; Wong, Y. S.; Hussaini, M. Y.: Spectral multigrid methods for elliptic equations I. J. comput. Phys. 48, 485-501 (1982) · Zbl 0496.65061
[50] Zang, T. A.; Wong, Y. S.; Hussaini, M. Y.: Spectral multigrid methods for elliptic equations II. J. comput. Phys. 54, 489-507 (1984) · Zbl 0543.65071