The discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region can be treated as modifications to matrix problems with very special structure. We show how to use the direct method of matrix decomposition to formulate an effective numerical algorithm for these problems. For typical applications the operation count is $O(N^3 )$ for an $N \times N$ grid. Numerical comparisons with other techniques are included.

The Direct Solution of the Biharmonic Equation on Rectangular Regions and the Poisson Equation on Irregular Regions

Direct solution of the biharmonic equation using noncoupled approach

A spectral-element method based on the Legendre–Galerkin approximation is presented to solve the two-dimensional biharmonic equations. Rigorous error analysis is carried out to establish the convergence of the method. By constructing appropriate basis functions which satisfy the boundary conditions of the differential equations, the discrete variational formulation is reduced to linear system with sparse and symmetric matrices, which can be efficiently solved by a fast Schur-complement method. Accuracy test is provided to confirm the convergence rate of the theoretical results. Finally, the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero Reynolds number flow in a driven cavity, the results compare well with established results.

Legendre–Galerkin spectral-element method for the biharmonic equations and its applications

A study of the flow of viscous liquids contained in an open cavity driven by a moving wall has been undertaken as a first step toward understanding interphase heat and mass transport in polymer processing equipment where rolling pools of liquid are generated. Solutions to the equations of motion for creeping flow in cavities of different aspect ratios were obtained using a finite-differences technique for circumstances when the wall moves in the same direction as the gravitational field.

Experimental studies of the flow patterns in cavities of various aspect ratios were found to be in good agreement with the creeping flow predictions for cavity Reynolds numbers Re < 50 and aspect ratios a < 1. The structure of the flow in this geometry was found to be similar to that in closed cavities of the same shape. The position and strength of the principal vortex were observed to be nearly independent of the existence of a free surface, and the gas-liquid interface was found to significantly stabilize the rotational motion in shallow cavities.

Flow in driven cavities with a free surface

This paper presents a fast iterative algorithm for the solution of a finite difference approximation of the biharmonic boundary value problem on a rectangular region. For solving this problem, the matrix decomposition algorithm is efficiently applied to the semi-direct method which essentially treats the biharmonic equation as a coupled system of Poisson equations. Assuming anN×N grid of mesh points, the number of operations required for one iteration and for the solution terminated by 0 (N
−2) is 0 (N
2) and 0 (N
5/2 log2
N), respectively. ForN
2 processors, the parallel version of this algorithm would require 14 log2
N steps per iteration. Both results are better than those known. A numerical experiment in a serial computation is also given.

A fast algorithm for solving the first biharmonic boundary value problem

Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems. The equation is treated as a system of two second order equations and a simple smoothing process is essential for convergence. An application is made to a crack-type problem.

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Fast finite-difference solution of biharmonic problems

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Fast finite-difference solution of biharmonic problems