An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem.

A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves fast Fourier transforms and, in comparison to its one-step counterpart, it has the advantage of requiring the solution a symmetric positive definite Schur complement system rather than a nonsymmetric one. As a consequence, the corresponding step of the new method is performed using a preconditioned conjugate gradient method. The total cost of the method on a N × N partition of the unit square is O (N 2 log N) . To demonstrate the optimal accuracy of the method, the results of numerical experiments are provided. [ABSTRACT FROM AUTHOR]