1549301742-The_Theory_of_Difference_Schemes__Samarskii

140 Basic Concepts of the Theory of Difference Schemes

In just the same way as before, we write down the boundary condition in

(9) for i = N:

aN+1(YN+1 -yN )-(yN -yN-1)

h2 'f!N l

In such a setting problem (9) turns out to be equivalent to the first boun-

dary-value problem

(58) i=O,l, ... ,N,

where <pi= t.p;, i = 1,2,.. .,N-1, (/; 0 = ~<p 0 , <pN = ~'PN, ai = 1,

i = 1,2, ... ,N and a 0 = hcru aN+i = hcr 2. If y is a solution of problem

(9), then

N-1

[A-^1 <p,<p]=[y,<p]=~h(y 0 t.pa+YN'PN)+ L Yi'Pih

i=l

N-1 N

L Y;'P;h+h(yotf>o+YNtpN)= L Yitpih

i=l i=O

N

L (A-l<p )i <pi h.

i=O

By applying successively formulae (51)-(52) to problern (58) and estimate

(53) we arrive at (55) and (57).

Corollary If cr 2 > c 1 > 0, then operator (54) admits the estimate

(59)

6. Operator equations of divergent type. We now deal with operators of
   the special structure known as divergent or conservative operators:

(60) A= T*ST,

where T, S and T* are linear bounded operators. Operators of this type
are frequently encountered in this book in the approximation of differential
operators having the form Lu = div ( k grad u). Let H be a vector space