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.

INTRODUCTION

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.