A fast direct solver for elliptic problems with a divergence constraint

Abstract

A fast direct solution method for a discretized vector-valued elliptic partial differential equation with a divergence constraint is considered. Such problems are typical in many disciplines such as fluid dynamics, elasticity and electromagnetics. The method requires the problem to be posed in a rectangle and boundary conditions to be either periodic boundary conditions or the so-called slip boundary conditions in one co-ordinate direction. The arising saddle-point matrix has a separable form when bilinear finite elements are used in the discretization. Based on a result for so-called p-circulant matrices, the saddle-point matrix can be transformed into a block-diagonal form by fast Fourier transformations. Thus, the fast direct solver has the same structure as methods for scalar-valued problems which are based on Fourier analysis and, therefore, it has the same computational cost ��(N log N). Numerical experiments demonstrate the good efficiency and accuracy of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.