1549301742-The_Theory_of_Difference_Schemes__Samarskii

118 Basic Concepts of the Theory of Difference Schemes

The right-hand side vector<[>= (<p 1 , <p 2 , .•• , <pN_ 1 ) includes the right-hand

sides of the boundary conditions (3)

<p 1 = f; , i = 2, ... , N - 2,

·u

<f'N-1 = f N-1 + h~ >

so that 'Pi may differ from f; only at the grid nodes adjacent to the boundary

points, that is, for i = 1 and i = N - 1.

The matrix U specifies an operator A = -A, carrying a grid function

y( xi), that is, a vector of the ( N - 1 )-dimensional space into a vector of

the same space (into the grid function (-A y);). The operator A coincides

with the operator A on all grid functions vanishing at the boundary nodes

(for i = 0 and i = N), so that (Ay); = (A y)i for i = 2, 3, ... , N - 2 and

(4)

Let Qh be the set of grid functions defined at the inner nodes of the
grid wh. The set so constructed is certainly linear. Once equipped with the
inner product (y, v) = L;:,_~^1 Yi vi hand associated norm II Yll = ~,
the space Qh becomes a normed vector space. The above operator A is
linear and maps Qh into itself, meaning that the domain and range of the
operator A coincide with the entire space Qh.
0
Let Qh be the space of all grid functions defined for all nodes of the
grid w h and vanishing at the boundary nodes, that is, for x E 11 ,. Then
0
the operator A may be treated as an operator from rt into Qh. Obviously,
A = Ah, y = Yh and <p = 'Ph depend solely on the grid step h. Just for
this reason the subject of subsequent discussions is a family of equations
depending on the parameter h rather than the single equation (2). A family
of such equations constitutes what is called an operator-difference scheme
(see Section 1.2).
While solving the operator equations (2) we establish the basic prop-
erties of the operator A such as self-adjointness, positive definiteness, the
lower bound of the operator and its norm and more. The operator A con-
structed in Example 1 will be frequently encountered in the sequel. Before
stating the main results, will be sensible to list its basic properties.
The operator A is self-adjoint, that is, (Ay, v) = (y, Av) for any
y, v E nh. As a matter of fact, (Ay, v) = (-A y, v). Making use of the
second Green formula (Section 3) and taking into account that A coincides
with A on the set of grid functions vanishing at the boundary nodes, we
establish the relation (Ay, v) = (y, Av), implying that A= A*.