A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem

Abstract

Abstract.A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a preconditioned conjugate gradient method. The total cost of the algorithm is O (N 3 ). Numerical results demonstrate the spectral convergence of the method.Mathematics Subject Classification. 65N35, 65N22.

Stokes eigenmodes in square domain and the stream function–vorticity correlation

“…Since the early work of Taylor [35] on the leading eigenmode of the buckling load problem, only a few attempts have been made to compute few Stokes eigenvalues and/or eigenmodes in fully confined geometries [5][6][7][8][9][10][11]13,[36][37][38]. Apart from the theoretical predictions proposed in [14] on the asymptotic behavior…”

Matrix decomposition algorithms for elliptic boundary value problems: a survey

“…A similar idea to that developed in [40] is used in [38], where a Legendre spectral collocation method is formulated for the solution of the mixed form of the biharmonic Dirichlet problem (3.23), (3.26) on a square. The solution and its Laplacian are approximated using the basis functions given by (6.3).…”

“…The work on the biharmonic Dirichlet problem described in [38,39] is related to MDAs in [50] and [176] for a Legendre spectral Galerkin method applied to the biharmonic equation directly instead of the mixed formulation considered in [39]. Legendre spectral Galerkin MDAs are used in [10] and [11] for the solution of the two-and three-dimensional Helmholtz equations, respectively.…”

A Legendre Petrov–Galerkin method for fourth-order differential equations

“…In computation, if we choose {φ n } as the basis of V N and take v = ψ m for 0 ≤ m ≤ N − 4, the linear part in the scheme can be written in a matrix form as in (4). The nonlinear term can be computed by the fast Legendre transform between the coefficients of the Legendre series and its values at the CGL points, such as…”

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