Fast numerical solution of the biharmonic Dirichlet problem on rectangles. (English) Zbl 0561.65077

A new method for the numerical solution of the first biharmonic Dirichlet problem in a rectangular domain is presented. For an \(N\times N\) mesh the complexity of this algorithm is on the order of \(N^ 2\) arithmetic operations. Only one array of order \(N^ 2\) 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 \(\epsilon\) after at most \(k=\ell n(2/\epsilon)\) 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.


65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J40 Boundary value problems for higher-order elliptic equations
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
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