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A fast direct solver for elliptic problems with a divergence constraint
Author(s) -
Martikainen Janne,
Rossi Tuomo,
Toivanen Jari
Publication year - 2002
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.283
Subject(s) - mathematics , discretization , circulant matrix , solver , partial differential equation , boundary value problem , mathematical analysis , saddle point , mathematical optimization , geometry , algorithm
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.