Stokes eigenmodes in square domain and the stream function-vorticity correlation
Journal of Computational Physics
PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform
Journal of Mathematical Imaging and Vision
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
Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations
SIAM Journal on Scientific Computing
A fourth order finite difference method for the Dirichlet biharmonic problem
Numerical Algorithms
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A Legendre spectral Galerkin 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 two 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 or the Cholesky method. The total cost of the algorithm is O(N3). Numerical results demonstrate the spectral convergence of the method.