Fast Numerical Solution of the Biharmonic Dirichlet Problem on Rectangles

Abstract

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

Preconditioning a class of fourth order problems by operator splitting

“…We refer to [38] as well as [15,16,19,44,45] for this case, which is important in plate theory but is excluded from the present analysis.…”

“…In this paper we develop preconditioners for spatially discretized versions of this system, for instance via finite element methods, thereby relying on the existence of efficient methods and software for each component of the system [18,20,21,23,37,40,48,49,50,52,53,54], especially on graded meshes [1,26,49,51]. Similar ideas have been developed for the bi-harmonic operator with Dirichlet boundary conditions u = ∂ ν u = 0, typical in structural mechanics, and quasi-uniform meshes in [15,16,19,44,45]. Materials science and fluid dynamics problems come with different boundary conditions which yield the operator splitting (1.1).…”

On the numerical solution of the first biharmonic equation

“…The preconditioning by the block diagonal part diag (L n , L 22 ) is investigated. Each block L u corresponds to the biharmonic problem with Au rather than u n specified on two opposite sides of the square ft. Bjjdrstad [2] has observed that this problem is easy to solve, since séparation of the variables is possible. Assuming that Au is specified at the left and right part of the boundary, we choose a row-wise ordering of the …”

“…The second member of the right hand side of (7.11) can be estimated using LEMMA 7.4 ; Let v (2) = (t? 0 , 0 Inserting (7.14) into (7.13) complètes the proof.…”

“…Following this idea we will provide a preconditioning matrix C h , such that the resulting condition number becomes independent of the mesh size. The matrix C h is based on the inverse of the square root of a discretization of -d 2 /ds 2 with homogeneous boundary conditions on each line segment v k of an.…”