Garrett Birkhoff , Richard S. Varga, and David Young, Alternating direction implicit methods, Advances in Computers 3 (F. Alt, ed. ), pp. 189-273, Academic Press, Inc., New York, 1962. MR 29, 5395. Zbl. 111, p. 314.
1. General Remarks
Alternating direction implicit methods, or ADI methods as they are called for short, consitute powerful techniques for solving elliptic and parabolic partial difference equations. However, in contrast with systematic overrelaxation methods, their effectiveness is hard to explain rigorously with any generality. Indeed, to provide a rational explanation for their effectiveness, must be regarded as a major unsolved problem of linear numerical analysis.
The present article attempts to survey the current status of this problem, as regards elliptic partial difference equation in the plane. It is divided into four chapters and four appendices.