Collocation and full multigrid methods
Applied Mathematics and Computation
Line relaxation for spectral multigrid methods
Journal of Computational Physics
SIAM Journal on Scientific and Statistical Computing
Spectral collocation methods for Stokes flow in contraction geometries and unbounded domains
Journal of Computational Physics
A fast algorithm for the evaluation of Legendre expansions
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
An efficient spectral method for ordinary differential equations with rational function coefficients
Mathematics of Computation
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem
SIAM Journal on Scientific Computing
Mathematics and Computers in Simulation
A Legendre Petrov-Galerkin method for fourth-order differential equations
Computers & Mathematics with Applications
Matrix decomposition algorithms for elliptic boundary value problems: a survey
Numerical Algorithms
Computers & Mathematics with Applications
A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations
Mathematical and Computer Modelling: An International Journal
Computers & Mathematics with Applications
Journal of Computational Physics
Applied Numerical Mathematics
Journal of Computational Physics
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It is well known that for the discretization of the biharmonic operator with spectral methods (Galerkin, tau, or collocation) we have a condition number of O(N^8), where N is the number of retained modes of approximations. This paper presents some efficient spectral algorithms, for reducing this condition number to O(N^4), based on the Jacobi-Galerkin methods for fourth-order equations in one variable. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. Jacobi-Galerkin methods for fourth-order equations in two dimension are considered. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate at large N values than that based on the Chebyshev- and Legendre-Galerkin methods.