A fourth order finite difference method for the Dirichlet biharmonic problem

Using the coupled approach, we formulate a fourth order finite difference scheme for the solution of the Dirichlet biharmonic problem on the unit square. On an N × N uniform partition of the square the scheme is solved at a cost O(N ² log₂ N)+m8N ² using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. Numerical tests confirm the fourth order accuracy of the scheme at the partition nodes with m proportional to log₂ N.