Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

doi:10.1016/j.apnum.2007.07.001

Applied Numerical Mathematics

Volume 58, Issue 8, August 2008, Pages 1224-1244

https://doi.org/10.1016/j.apnum.2007.07.001 Get rights and content

Abstract

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.

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Citation Excerpt :
The Stokes and Navier–Stokes equations involving the stream-function formulation were solved by spectral collocation-type methods in [3]. Based on the Jacobi–Galerkin methods, [7] has presented some efficient direct solvers for general fourth-order equations subject to various boundary conditions. In the mixed approach [8–16], the fourth-order equation is first replaced by a coupled system of two second-order differential equations, and this system is then discretized by the finite difference or finite element method.
A spectral-element method based on the Legendre–Galerkin approximation is presented to solve the two-dimensional biharmonic equations. Rigorous error analysis is carried out to establish the convergence of the method. By constructing appropriate basis functions which satisfy the boundary conditions of the differential equations, the discrete variational formulation is reduced to linear system with sparse and symmetric matrices, which can be efficiently solved by a fast Schur-complement method. Accuracy test is provided to confirm the convergence rate of the theoretical results. Finally, the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero Reynolds number flow in a driven cavity, the results compare well with established results.
A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems
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Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

Abstract

Comput. & Structures

J. Comput. Math. Appl.

J. Comput. Appl. Math.

J. Math. Anal. Appl.

J. Comput. Phys.

Appl. Math. Comput.

J. Comput. Phys.

Acta Astronaut.

J. Math. Anal. Appl.

J. Comput. Phys.

J. Comput. Phys.

A fast algorithm for the evaluation of Legendre expansions

SIAM J. Sci. Statist. Comput.

Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions

Numer. Math.

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.

Some spectral approximations of fourth-order problems

A Legendre spectral collocation method for the biharmonic Dirichlet problem

Math. Model. Numer. Anal.

A Legendre spectral Galerkin method for the biharmonic Dirichlet problem

SIAM J. Sci. Comput.

Fast numerical solution of the biharmonic Dirichlet problem on rectangles

SIAM J. Numer. Anal.

Efficient algorithms for solving a fourth-order equation with the spectral-Galerkin method

SIAM J. Sci. Comput.

Chebyshev and Fourier Spectral Methods

The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular domains

SIAM J. Numer. Anal.

Spectral Methods in Fluid Dynamics

A mixed finite element method for the biharmonic equation