Fast Numerical Solution of the Biharmonic Dirichlet Problem on Rectangles

A new method for the numerical solution of the first biharmonic Dirichlet problem in a rectangular domain is presented. For an N × N mesh the complexity of this algorithm is on the order of N² arithmetic operations. Only one array of order N × N mesh the complexity of this algorithm is on the order of N² and a workspace of size less than $10N$ are required. These results are therefore optimal and the algorithm is an order of magnitude more efficient than previously known methods with the possible exception of multi-grid. The method has an iterative part where a problem with different boundary conditions is used to precondition the original problem. It is shown that any initial error will be reduced by a factor ε after at most K = ln (2 / ε ) iterations using the conjugate gradient method. The conjugate gradient method is also shown to have a superlinear rate of convergence when applied to this formulation of the problem. The purpose of this paper is to provide a description and analysis of the new method.