1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General forinulations 127

Stability of the scheme is to be understood as the property that the inverse

Ah^1 acting from B 1 ~^2 ) into B~^1 ) is uniformly bounded in h:

(24) II Ah^1 II < M where M > 0 does not depend on h.

Combination of (23) and (24) gives estimate (22):

II Yh ll(h) <II Ahl II. II 'Ph ll(h) < M II 'Ph ll(h)'

That is to say, the meaning of stability of scheme (21) is that a solution (21)

depends continuously on the right-hand side and this dependence is uniform

in the parameter h. This implies that a small change of the right-hand side

results in a small change of the solution. If the scheme is solvable and

stable, it is correct. Note that the uniqueness of the scheme (21) solution

is a consequence of its solvability and stability and, hence, we might get

rid of the uniqueness requirement in condition (1). Indeed, assume to the

contrary that there were two solutions to equation (21), say ·fh and yh f- ·fh.

By the linearity property of the operator Ah, their difference zh = yh - ·fh

should satisfy the homogeneous equation

Ah zh =Ah Cfh - Yh) =Ah 'fh - Ah Yh ='Ph - 'Ph= 0 ·

Because scheme (21) is stable, inequality (22) holds true and, therefore,

implying that Yh = y 1 ,.

To prove the stability of (21), we need an a priori estimate of the form

(22). A derivation of some a priori estimates for the operator equation (21)

will be carried out in Section 4. A difference scheme Ah Yh = 'Ph is said

to be ill-posed if at least one of the conditions (1)-(2) we have imposed

above is violated..

Suppose that a solution Yh of problem (21) exists for any 'Ph E B,;^2 l, so

that yh^1 = Ah^1 'Ph·· Since B~^1 ) and B~^2 ) are finite-dimensional spaces, the

inverse operator Ah^1 from B,;^2 ) into B~^1 ) is bounded and its norm equals
II Ah^1 II= j\lfh, where Mh is a positive constant depending on the parameter
h. If the scheme is stable, there exists a constant M > 0 independent of h
such that M h < M for all I h I < h 0. The meaning of instability of scheme
(21) and, hence, of its ill-posedness is that Mh ---+ oo as I h I---+ 0, it being
understood that the above constant M does not exist. For an ill-posed
problem only an estimate of the form

(25)