Basic Concepts of the Theory
of Difference Schemes
In this chapter elementary examples illustrate the basic concepts of the the-
ory of difference sche1nes: approximation, stability and convergence. The
reader is already familiar with several 1nethods available for investigating
stability and convergence such as the method of separation of variables and
the method of energy inequalities. In Section 4 difference equations are
treated as operator equations in an abstract space. These provide a means
of studying a wide range of interesting methods in a unified manner. We
begin our exposition with a discussion of examples that make it possible to
draw fairly accurate outlines of the possible theory regarding these ques-
tions and with a listing of the basic results of the book together with a
development desired for them.
2.1 RELEVANT ELEMENTS OF FUNCTIONAL ANALYSISOur account of the theory of difference schen1es is mostly based on ele-
mentary notions from functional analysis. In what follows we list briefly
widespread tools adopted in the theory of linear operators which will be
used in the body of this book.
- Linear operators. Let X and Y be normed vector spaces and D be a
subspace of the space X. If to each vector x E D there corresponds by an
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