A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations
Journal of Scientific Computing
Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System
Journal of Scientific Computing
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 Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square
Journal of Scientific Computing
Journal of Scientific Computing
A fourth order finite difference method for the Dirichlet biharmonic problem
Numerical Algorithms
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We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [J. W. Stephenson, J. Comput. Phys., 55 (1984), pp. 65-80] as well as in the case of a new fourth order scheme, which is discussed in this paper and is based on the capacitance matrix method ([B. L. Buzbee and F. W. Dorr, SIAM J. Numer. Anal., 11 (1974), pp. 1136-1150], [P. Bjørstad, SIAM J. Numer. Anal., 20 (1983), pp. 59-71]). The discrete biharmonic operator is decomposed into two components. The first is a diagonal operator in the eigenfunction basis of the Laplacian, to which the FFT algorithm is applied. The second is a low-rank perturbation operator (given by the capacitance matrix), which is due to the deviation of the discrete operators from diagonal form. The Sherman-Morrison formula [G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, MD, 1996] is applied to obtain a fast solution of the resulting linear system of equations.