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

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(N8), where N is the number of retained modes of approximations. This paper presents some efficient spectral algorithms, for reducing this condition number to O(N4), 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.