1549301742-The_Theory_of_Difference_Schemes__Samarskii

Stability with respect to coefficients 233

via perturbations of I and A.

Suppose that the inverse operators A-^1 and JI-^1 exist. Moreover, we

assume that A and A are self-adjoint positive operators. Substitutions of

u = A-^11 and u = JI-^11 into (15) yield

Applying A^112 to both sides of equality (16) we find that

We will estimate the vector z in the norm II z llx = V(Az, z) of the

space HA together with the san1e things for I and J in the negative norm

II I llx-1 = V(A-^1 I, I) of the energy space H_4_ 1 • Via transform

we obtain

As a measure of perturbations of the operator A we adopt a relative vari-

ation of the energy (Ax, x) of the operator A. The meaning of this is that

we should have for all x E H

( 18) I ( (A - A) x, x) I < ex (Ax, x), ex> 0,

whence it follows that

( 19) (1 - ex) A< A< (1 +ex) A,

(20) (1-ex)A-^1 < A-^1 < (1 + ex)JI-^1.

Observe that (20) is an immediate implication of (19). To make sure

of it, we compose the difference J = (1 + ex)(Ax, x) - (Ax, x) and insert
A if2:c = y: