Preface XlllAn estimate of the convergence rate for the method of minimal cor-
rections is derived in the case when A is a non-self-adjoint operator and in
others situations.
Also, we consider the total approxi1nation method as a constructive
method for creating economical difference sche1nes for the multidimensional
equations of mathematical physics, The notion of additive scheme is intro-
duced as a system of operator difference equations that approximates the
original differential equation in the total sense. Two quite general heuristic
methods (proposed earlier by the author) for obtaining additive economical
schen1es are discussed in full details. The additive sche1nes require a new
technique for investigating convergence and a new type of a priori esti1nates
that take into account the definition of the property of approxin1ation.
We have not had the chance to discuss some works on difference 1neth-
ods of an applied character, although such works best illustrate the real pos-
sibilities of difference methods and provide constant sources of stinrnlation
for the formulation of new proble1ns.
vVhen eco1101nical schen1es for 1nulticlin1ensional problems in mathe-
matical physics are developed in Chapter 9, we shall need a revised concept
of approximation error, thereby changing the definition of sche1ne. The
notion of summed (in t) approximation in Section 3 of Chapter 9 is of a
constructive nature, making it possible to produce economical schemes for
various problems.
Last, but not least, I wish to thank those who have assisted me in
this enterprise.
Alexander Saniarskil